+
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\begin{abstract}
Most of the time, asynchronous load balancing algorithms have extensively been
-studied in a theoretical point of view. The Bertsekas and Tsitsiklis' algorithm
+studied in a theoretical point of view. The Bertsekas and Tsitsiklis'
+algorithm~\cite[section~7.4]{bertsekas+tsitsiklis.1997.parallel}
is certainly the most well known algorithm for which the convergence proof is
given. From a practical point of view, when a node wants to balance a part of
its load to some of its neighbors, the strategy is not described. In this
\end{abstract}
-
+\section{Introduction}
Load balancing algorithms are extensively used in parallel and distributed
applications in order to reduce the execution times. They can be applied in
proved that under classical hypotheses of asynchronous iterative algorithms and
a special constraint avoiding \texttt{ping-pong} effect, an asynchronous
iterative algorithm converge to the uniform load distribution. This work has
-been extended by many authors. For example, DASUD propose a version working with
-integer load.
+been extended by many authors. For example,
+DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous} propose a version working
+with integer load. {\bf 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
+the load distribution. In other word, a node can send a part of its load to one
+or many of its neighbors while all the convergence conditions are
+followed. Consequently, we propose a new strategy called \texttt{best effort}
+that tries to balance the load of a node to all its less loaded neighbors while
+ensuring that all the nodes concerned by the load balancing phase have the same
+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
+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
+message at each neighbor at each iteration. Latter messages contains data that
+migrates from one node to another one. Depending on the application, it may have
+sense or not that nodes try to balance a part of their load at each computing
+iteration. But the time to transfer a load message from a node to another one is
+often much nore longer that to time to transfer a load information message. So,
+when a node receives the information that later it will receive a data message,
+it can take this information into account and it can consider that its new load
+is larger. Consequently, it can send a part of it real load to some of its
+neighbors if required. We call this trick the \texttt{virtual load} mecanism.
+
+
+
+So, in this work, we propose a new strategy for improving the distribution of
+the load and a simple but efficient trick that also improves the load
+balacing. Moreover, we have conducted many simulations with simgrid in order to
+validate our improvements are really efficient. Our simulations consider that in
+order to send a message, a latency delays the sending and according to the
+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 mecanism 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.
+
+
+
+
+\section{Bertsekas and Tsitsiklis' asynchronous load balancing algorithm}
+\label{BT algo}
+
+In order prove the convergence of asynchronous iterative load balancing
+Bertesekas and Tsitsiklis proposed a model
+in~\cite{bertsekas+tsitsiklis.1997.parallel}. Here we recall some notations.
+Consider that $N={1,...,n}$ processors are connected through a network.
+Communication links are represented by a connected undirected graph $G=(N,V)$
+where $V$ is the set of links connecting differents processors. In this work, we
+consider that processors are homogeneous for sake of simplicity. It is quite
+easy to tackle the heterogeneous case~\cite{ElsMonPre02}. Load of processor $i$
+at time $t$ is represented by $x_i(t)\geq 0$. Let $V(i)$ be the set of
+neighbors of processor $i$. Each processor $i$ has an estimate of the load of
+each of its neighbors $j \in V(i)$ represented by $x_j^i(t)$. According to
+asynchronism and communication delays, this estimate may be outdated. We also
+consider that the load is described by a continuous variable.
+
+When a processor send a part of its load to one or some of its neighbors, the
+transfer takes time to be completed. Let $s_{ij}(t)$ be the amount of load that
+processor $i$ has transfered to processor $j$ at time $t$ and let $r_{ij}(t)$ be the
+amount of load received by processor $j$ from processor $i$ at time $t$. Then
+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}
+\end{equation}
+
+
+Some conditions are required to ensure the convergence. One of them can be
+called the \texttt{ping-pong} condition which specifies that:
+\begin{equation}
+x_i(t)-\sum _{k\in V(i)} s_{ik}(t) \geq x_j^i(t)+s_{ij}(t)
+\end{equation}
+for any processor $i$ and any $j \in V(i)$ such that $x_i(t)>x_j^i(t)$. This
+condition aims at avoiding a processor to send a part of its load and beeing
+less loaded after that.
+
+Nevertheless, we think that this condition may lead to deadlocks in some
+cases. For example, if we consider only three processors and that processor $1$
+is linked to processor $2$ which is also linked to processor $3$ (i.e. a simple
+chain wich 3 processors). Now consider we have the following values at time $t$:
+\begin{eqnarray*}
+x_1(t)=10 \\
+x_2(t)=100 \\
+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
+$x_3^2(t)$. So we consider that the \texttt{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.
+
+
+\section{Best effort strategy}
+\label{Best-effort}
+
+
+
+\section{Virtual load}
+\label{Virtual load}
+
+\section{Simulations}
+\label{Simulations}
+
+In order to test and validate our approaches, we wrote a simulator
+using the SimGrid
+framework~\cite{casanova+legrand+quinson.2008.simgrid}. The process
+model is detailed in the next section (\ref{Sim model}), then the
+results of the simulations are presented in section~\ref{Results}.
+
+\subsection{Simulation model}
+\label{Sim model}
+
+\subsection{Validation of our approaches}
+\label{Results}
+
+
+On veut montrer quoi ? :
+
+1) best plus rapide que les autres (simple, makhoul)
+2) avantage virtual load
+
+Est ce qu'on peut trouver des contre exemple?
+Topologies variées
+
+
+Simulation avec temps définies assez long et on mesure la qualité avec : volume de calcul effectué, volume de données échangées
+Mais aussi simulation avec temps court qui montre que seul best converge
+
+
+Expés avec ratio calcul/comm rapide et lent
+
+Quelques expés avec charge initiale aléatoire plutot que sur le premier proc
+
+Cadre processeurs homogènes
+
+Topologies statiques
+
+On ne tient pas compte de la vitesse des liens donc on la considère homogène
+
+Prendre un réseau hétérogène et rendre processeur homogène
+
+Taille : 10 100 très gros
+
+\section{Conclusion and perspectives}
+\bibliographystyle{spmpsci}
+\bibliography{biblio}
\end{document}
