+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+asynchronism and communication delays, this estimate may be outdated.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.
+
