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,
+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} mecanism.
+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
-balacing. Moreover, we have conducted many simulations with simgrid in order to
+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
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
+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.
\label{BT algo}
In order prove the convergence of asynchronous iterative load balancing
-Bertesekas and Tsitsiklis proposed a model
+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 differents processors. In this work, we
+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
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
+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)-\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
+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 wich 3 processors). Now consider we have the following values at time $t$:
+chain which 3 processors). Now consider we have the following values at time $t$:
\begin{eqnarray*}
x_1(t)=10 \\
x_2(t)=100 \\
%%% ispell-local-dictionary: "american"
%%% End:
-% LocalWords: Raphaël Couturier Arnaud Giersch Abderrahmane Sider
-% LocalWords: Bertsekas Tsitsiklis SimGrid DASUD
+% LocalWords: Raphaël Couturier Arnaud Giersch Abderrahmane Sider Franche ij
+% LocalWords: Bertsekas Tsitsiklis SimGrid DASUD Comté Béjaïa asynchronism ji
+% LocalWords: ik
