+\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
+different scientific fields from high performance computation to micro sensor
+networks. They are iterative by nature. In literature many kinds of load
+balancing algorithms have been studied. They can be classified according
+different criteria: centralized or decentralized, in static or dynamic
+environment, with homogeneous or heterogeneous load, using synchronous or
+asynchronous iterations, with a static topology or a dynamic one which evolves
+during time. In this work, we focus on asynchronous load balancing algorithms
+where computer nodes are considered homogeneous and with homogeneous load with
+no external load. In this context, Bertsekas and Tsitsiklis have proposed an
+algorithm which is definitively a reference for many works. In their work, they
+proved that under classical hypotheses of asynchronous iterative algorithms and
+a special constraint avoiding \emph{ping-pong} effect, an asynchronous
+iterative algorithm converge to the uniform load distribution. This work has
+been extended by many authors. For example, Cortés et al., with
+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}.
+\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
+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 \emph{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. 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
+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 more 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 \emph{virtual load} mechanism.
+
+
+
+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
+balancing. 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. 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.
+
+
+
+\section{Bertsekas and Tsitsiklis' asynchronous load balancing algorithm}
+\label{BT algo}
+
+In order prove the convergence of asynchronous iterative load balancing
+Bertsekas 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 different 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 transferred 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 \emph{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 being
+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 which 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 \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}
+
+In this section we describe a new load-balancing strategy that we call
+\emph{best effort}. The general idea behind this strategy is that each
+processor, that detects it has more load than some of its neighbors,
+sends some load to the most of its less loaded neighbors, doing its
+best to reach the equilibrium between those neighbors and himself.
+
+More precisely, when a processor $i$ is in its load-balancing phase,
+he proceeds as following.
+\begin{enumerate}
+\item First, the neighbors are sorted in non-decreasing order of their
+ known loads $x^i_j(t)$.
+
+\item Then, this sorted list is traversed in order to find its largest
+ prefix such as the load of each selected neighbor is lesser than:
+ \begin{itemize}
+ \item the processor's own load, and
+ \item the mean of the loads of the selected neighbors and of the
+ processor's load.
+ \end{itemize}
+ Let's call $S_i(t)$ the set of the selected neighbors, and
+ $\bar{x}(t)$ the mean of the loads of the selected neighbors and of
+ the processor load:
+ \begin{equation*}
+ \bar{x}(t) = \frac{1}{\abs{S_i(t)} + 1}
+ \left( x_i(t) + \sum_{j\in S_i(t)} x^i_j(t) \right)
+ \end{equation*}
+ The following properties hold:
+ \begin{equation*}
+ \begin{cases}
+ S_i(t) \subset V(i) \\
+ x^i_j(t) < x_i(t) & \forall j \in S_i(t) \\
+ x^i_j(t) < \bar{x} & \forall j \in S_i(t) \\
+ x^i_j(t) \leq x^i_k(t) & \forall j \in S_i(t), \forall k \in V(i) \setminus S_i(t) \\
+ \bar{x} \leq x_i(t)
+ \end{cases}
+ \end{equation*}
+
+\item Once this selection is completed, processor $i$ sends to each of
+ the selected neighbor $j\in S_i(t)$ an amount of load $s_{ij}(t) =
+ \bar{x} - x^i_j(t)$.
+
+ From the above equations, and notably from the definition of
+ $\bar{x}$, it can easily be verified that:
+ \begin{equation*}
+ \begin{cases}
+ x_i(t) - \sum_{j\in S_i(t)} s_{ij}(t) = \bar{x} \\
+ x^i_j(t) + s_{ij}(t) = \bar{x} & \forall j \in S_i(t)
+ \end{cases}
+ \end{equation*}
+\end{enumerate}
+
+\FIXME{describe parameter $k$}
+
+\section{Other strategies}
+\label{Other}
+
+\FIXME{Réécrire en angliche.}
+
+% \FIXME{faut-il décrire les stratégies makhoul et simple ?}
+
+% \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
+voisins.
+
+Ensuite, pour chaque voisin, dans l'ordre, et tant qu'on reste plus
+chargé que le voisin en question, on lui envoie 1/(N+1) de la
+différence calculée au départ, avec N le nombre de voisins.
+
+C'est l'algorithme~2 dans~\cite{bahi+giersch+makhoul.2008.scalable}.
+
+\section{Virtual load}
+\label{Virtual load}
+
+In this section, we present the concept of \texttt{virtual load}. In order to
+use this concept, load balancing messages must be sent using two different kinds
+of messages: load information messages and load balancing messages. More
+precisely, a node wanting to send a part of its load to one of its neighbors,
+can first send a load information message containing the load it will send and
+then it can send the load balancing message containing data to be transferred.
+Load information message are really short, consequently they will be received
+very quickly. In opposition, load balancing messages are often bigger and thus
+require more time to be transferred.
+
+The concept of \texttt{virtual load} allows a node that received a load
+information message to integrate the load that it will receive later in its load
+(virtually) and consequently send a (real) part of its load to some of its
+neighbors. In fact, a node that receives a load information message knows that
+later it will receive the corresponding load balancing message containing the
+corresponding data. So if this node detects it is too loaded compared to some
+of its neighbors and if it has enough load (real load), then it can send more
+load to some of its neighbors without waiting the reception of the load
+balancing message.
+
+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.
+
+\FIXME{Est ce qu'on donne l'algo avec virtual load?}
+
+\FIXME{describe integer mode}
+
+\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}. This
+simulator, which consists of about 2,700 lines of C++, allows to run
+the different load-balancing strategies under various parameters, such
+as the initial distribution of load, the interconnection topology, the
+characteristics of the running platform, etc. Then several metrics
+are issued that permit to compare the strategies.
