X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/1ccb8a3cdf8051ce59428e89b46322f8b6326db2..50f790a50dfec79e4aeabbe564c4c7d375c7d57f:/supercomp11/supercomp11.tex?ds=inline

diff --git a/supercomp11/supercomp11.tex b/supercomp11/supercomp11.tex
index 2fc63f7..4a7e57a 100644
--- a/supercomp11/supercomp11.tex
+++ b/supercomp11/supercomp11.tex
@@ -1,5 +1,8 @@
-
 \documentclass[smallextended]{svjour3}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{mathptmx}
+\usepackage{courier}
 \usepackage{graphicx}
 
 \begin{document}
@@ -28,7 +31,8 @@
 \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
@@ -40,7 +44,7 @@ balancing  algorithm is  implemented most  of the  time can  dissociate messages
 concerning load transfers and message  concerning load information.  In order to
 increase  the  converge of  a  load balancing  algorithm,  we  propose a  simple
 heuristic called \texttt{virtual load} which allows a node that receives an load
-information message  to integrate the  load that it  will receive latter  in its
+information message  to integrate the  load that it  will receive later  in its
 load (virtually) and consequently sends a (real) part of its load to some of its
 neighbors.  In order to  validate our  approaches, we  have defined  a simulator
 based on SimGrid which allowed us to conduct many experiments.
@@ -65,8 +69,9 @@ 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  \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 proposes a version working with
-integer load. {\bf Rajouter des choses ici}.
+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
@@ -117,7 +122,29 @@ conclusion and some perspectives to this work.
 \section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}
 \label{BT algo}
 
-Comment on the problem in the convergence condition.
+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)
+\end{equation}
+
 
 \section{Best effort strategy}
 \label{Best-effort}
@@ -135,8 +162,46 @@ Comment on the problem in the convergence condition.
 \subsection{Validation of our approaches}
 
 
+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 variÃ©es
+
+
+Simulation avec temps dÃ©finies assez long et on mesure la qualitÃ© avec : volume de calcul effectuÃ©, volume de donnÃ©es Ã©changÃ©es
+Mais aussi simulation avec temps court qui montre que seul best converge
+
+
+ExpÃ©s avec ratio calcul/comm rapide et lent
+
+Quelques expÃ©s avec charge initiale alÃ©atoire plutot que sur le premier proc
+
+Cadre processeurs homogÃ¨nes
+
+Topologies statiques
+
+On ne tient pas compte de la vitesse des liens donc on la considÃ¨re homogÃ¨ne
+
+Prendre un rÃ©seau hÃ©tÃ©rogÃ¨ne et rendre processeur homogÃ¨ne
+
+Taille : 10 100 trÃ¨s gros
+
 \section{Conclusion and perspectives}
 
 
+\bibliographystyle{spmpsci}
+\bibliography{biblio}
 
 \end{document}
+
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: t
+%%% ispell-local-dictionary: "american"
+%%% End:
+
+% LocalWords:  RaphaÃ«l Couturier Arnaud Giersch Abderrahmane Sider
+% LocalWords:  Bertsekas Tsitsiklis SimGrid DASUD
