X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/d883af6fcab703af629fd6ca368b7076b5b3384a..ed88631cce8e6737ba146dd0a98914baa97ee29c:/supercomp11/supercomp11.tex

diff --git a/supercomp11/supercomp11.tex b/supercomp11/supercomp11.tex
index d1ff213..a566b8a 100644
--- a/supercomp11/supercomp11.tex
+++ b/supercomp11/supercomp11.tex
@@ -1,3 +1,4 @@
+
 \documentclass[smallextended]{svjour3}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
@@ -122,12 +123,95 @@ 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+asynchronism and communication  delays, this estimate may be  outdated.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)
+\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}
 
+\textbf{À traduire} Ordonne les voisins du moins chargé au plus chargé.
+Trouve ensuite, en les prenant dans ce ordre, le nombre maximal de
+voisins tels que tous ont une charge inférieure à la moyenne des
+charges des voisins sélectionnés, et de soi-même.
+
+Les transferts de charge sont ensuite fait en visant cette moyenne pour
+tous les voisins sélectionnés.  On envoie une quantité de charge égale
+à (moyenne - charge\_du\_voisin).
+
+~\\\textbf{Question} 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}
@@ -135,9 +219,74 @@ Comment on the problem in the convergence condition.
 \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}
+
+\begin{verbatim}
+Communications
+==============
+
+There are two receiving channels per host: control for information
+messages, and data for load transfers.
+
+Process model
+=============
+
+Each process is made of 3 threads: a receiver thread, a computing
+thread, and a load-balancer thread.
+
+* Receiver thread
+  ---------------
+
+    Loop
+    | wait for a message to come, either on data channel, or on ctrl channel
+    | push received message in a buffer of received messages
+    | -> ctrl messages on the one side
+    | -> data messages on the other side
+    +-
+
+   The loop terminates when a "finalize" message is received on each
+   channel.
+
+* Computing thread
+  ----------------
+
+    Loop
+    | if we received some real load, get it (data messages)
+    | if there is some real load to send, send it
+    | if we own some load, simulate some computing on it
+    | sleep a bit if we are looping too fast
+    +-
+    send CLOSE on data for all neighbors
+    wait for CLOSE on data from all neighbors
+
+  The loop terminates when process::still_running() returns false.
+  (read the source for full details...)
+
+* Load-balancing thread
+  ---------------------
+
+    Loop
+    | call load-balancing algorithm
+    | send ctrl messages
+    | sleep (min_lb_iter_duration)
+    | receive ctrl messages
+    +-
+    send CLOSE on ctrl for all neighbors
+    wait for CLOSE on ctrl from all neighbors
+
+  The loop terminates when process::still_running() returns false.
+  (read the source for full details...)
+\end{verbatim}
 
 \subsection{Validation of our approaches}
+\label{Results}
 
 
 On veut montrer quoi ? :