X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/loba-papers.git/blobdiff_plain/d883af6fcab703af629fd6ca368b7076b5b3384a..3dde28334c43de22323538f3bab0c463fa4a5346:/supercomp11/supercomp11.tex

diff --git a/supercomp11/supercomp11.tex b/supercomp11/supercomp11.tex
index d1ff213..c6e904a 100644
--- a/supercomp11/supercomp11.tex
+++ b/supercomp11/supercomp11.tex
@@ -2,27 +2,42 @@
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{mathptmx}
+\usepackage{amsmath}
 \usepackage{courier}
 \usepackage{graphicx}
+\usepackage{url}
+\usepackage[ruled,lined]{algorithm2e}
+
+\newcommand{\abs}[1]{\lvert#1\rvert} % \abs{x} -> |x|
+
+\newenvironment{algodata}{%
+  \begin{tabular}[t]{@{}l@{:~}l@{}}}{%
+  \end{tabular}}
+
+\newcommand{\FIXMEmargin}[1]{%
+  \marginpar{\textbf{[FIXME]} {\footnotesize #1}}}
+\newcommand{\FIXME}[2][]{%
+  \ifx #2\relax\relax \FIXMEmargin{#1}%
+  \else \textbf{$\triangleright$\FIXMEmargin{#1}~#2}\fi}
+
+\newcommand{\VAR}[1]{\textit{#1}}
 
 \begin{document}
 
-\title{Best effort strategy and virtual load for asynchronous iterative load balancing}
+\title{Best effort strategy and virtual load
+  for asynchronous iterative load balancing}
 
 \author{Raphaël Couturier \and
-        Arnaud Giersch \and
-        Abderrahmane Sider
+        Arnaud Giersch
 }
 
-\institute{F. Author \at
-              first address \\
-              Tel.: +123-45-678910\\
-              Fax: +123-45-678910\\
-              \email{fauthor@example.com}           %  \\
-%             \emph{Present address:} of F. Author  %  if needed
-           \and
-           S. Author \at
-              second address
+\institute{R. Couturier \and A. Giersch \at
+              FEMTO-ST, University of Franche-Comté, Belfort, France \\
+              % Tel.: +123-45-678910\\
+              % Fax: +123-45-678910\\
+              \email{%
+                raphael.couturier@femto-st.fr,
+                arnaud.giersch@femto-st.fr}
 }
 
 \maketitle
@@ -36,14 +51,14 @@ 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
-paper, we propose a strategy  called \texttt{best effort} which tries to balance
+paper, we propose a strategy  called \emph{best effort} which 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,  asynchronous  iterative  algorithms  in which  an  asynchronous  load
 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
+heuristic called \emph{virtual load} which allows a node that receives a load
 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
@@ -67,23 +82,25 @@ 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  \texttt{ping-pong}  effect,  an  asynchronous
+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,
-DASUD~\cite{cortes+ripoll+cedo+al.2002.asynchronous} propose a version working
-with integer load. {\bf Rajouter des choses ici}.
+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.  Lesquelles ?}
 
 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  \texttt{best effort}
+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.  Formers  ones  allows  a  node to  inform  its
+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
@@ -91,56 +108,569 @@ 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 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 \texttt{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
 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. In Section~\ref{Virtual  load}, the  virtual load  mecanism 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.
-
+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}
 
-Comment on the problem in the convergence condition.
+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.
+
+Nevertheless, we conjecture that such a weaker condition exists.  In fact, we
+have never seen any scenario that is not leading to convergence, even with
+load-balancing strategies that are not exactly fulfilling these two conditions.
+
+It may be the subject of future work to express weaker conditions, and to prove
+that they are sufficient to ensure the convergence of the load-balancing
+algorithm.
 
 \section{Best effort strategy}
 \label{Best-effort}
 
+In this section we describe a new load-balancing strategy that we call
+\emph{best effort}.  First, we explain the general idea behind this strategy,
+and then we describe some variants of this basic strategy.
+
+\subsection{Basic strategy}
+
+The general idea behind the \emph{best effort} 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}
+
+\subsection{Leveling the amount to send}
+
+With the aforementioned basic strategy, each node does its best to reach the
+equilibrium with its neighbors.  Since each node may be taking the same kind of
+decision at the same moment, there is the risk that a node receives load from
+several of its neighbors, and then is temporary going off the equilibrium state.
+This is particularly true with strongly connected applications.
+
+In order to reduce this effect, we add the ability to level the amount to send.
+The idea, here, is to make smaller steps toward the equilibrium, such that a
+potentially wrong decision has a lower impact.
+
+Concretely, once $s_{ij}$ has been evaluated as before, it is simply divided by
+some configurable factor.  That's what we named the ``parameter $k$'' in
+Section~\ref{Results}.  The amount of data to send is then $s_{ij}(t) = (\bar{x}
+- x^i_j(t))/k$.
+\FIXME[check that it's still named $k$ in Sec.~\ref{Results}]{}
+
+\section{Other strategies}
+\label{Other}
+
+Another load balancing strategy, working under the same conditions, was
+previously developed by Bahi, Giersch, and Makhoul in
+\cite{bahi+giersch+makhoul.2008.scalable}.  In order to assess the performances
+of the new \emph{best effort}, we naturally chose to compare it to this anterior
+work.  More precisely, we will use the algorithm~2 from
+\cite{bahi+giersch+makhoul.2008.scalable} and, in the following, we will
+reference it under the name of Makhoul's.
+
+Here is an outline of the Makhoul's algorithm.  When a given node needs to take
+a load balancing decision, it starts by sorting its neighbors by increasing
+order of their load.  Then, it computes the difference between its own load, and
+the load of each of its neighbors.  Finally, taking the neighbors following the
+order defined before, the amount of load to send $s_{ij}$ is computed as
+$1/(N+1)$ of the load difference, with $N$ being the number of neighbors.  This
+process continues as long as the node is more loaded than the considered
+neighbor.
 
 
 \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.
+
+The simulation model is detailed in the next section (\ref{Sim
+  model}), and the experimental contexts are described in
+section~\ref{Contexts}.  Then the results of the simulations are
+presented in section~\ref{Results}.
+
 \subsection{Simulation model}
+\label{Sim model}
+
+In the simulation model the processors exchange messages which are of
+two kinds.  First, there are \emph{control messages} which only carry
+information that is exchanged between the processors, such as the
+current load, or the virtual load transfers if this option is
+selected.  These messages are rather small, and their size is
+constant.  Then, there are \emph{data messages} that carry the real
+load transferred between the processors.  The size of a data message
+is a function of the amount of load that it carries, and it can be
+pretty large.  In order to receive the messages, each processor has
+two receiving channels, one for each kind of messages.  Finally, when
+a message is sent or received, this is done by using the non-blocking
+primitives of SimGrid\footnote{That are \texttt{MSG\_task\_isend()},
+  and \texttt{MSG\_task\_irecv()}.}.
+
+During the simulation, each processor concurrently runs three threads:
+a \emph{receiving thread}, a \emph{computing thread}, and a
+\emph{load-balancing thread}, which we will briefly describe now.
+
+\paragraph{Receiving thread} The receiving thread is in charge of
+waiting for messages to come, either on the control channel, or on the
+data channel.  Its behavior is sketched by Algorithm~\ref{algo.recv}.
+When a message is received, it is pushed in a buffer of
+received message, to be later consumed by one of the other threads.
+There are two such buffers, one for the control messages, and one for
+the data messages.  The buffers are implemented with a lock-free FIFO
+\cite{sutter.2008.writing} to avoid contention between the threads.
+
+\begin{algorithm}
+  \caption{Receiving thread}
+  \label{algo.recv}
+  \KwData{
+    \begin{algodata}
+      \VAR{ctrl\_chan}, \VAR{data\_chan}
+      & communication channels (control and data) \\
+      \VAR{ctrl\_fifo}, \VAR{data\_fifo}
+      & buffers of received messages (control and data) \\
+    \end{algodata}}
+  \While{true}{%
+    wait for a message to be available on either \VAR{ctrl\_chan},
+    or \VAR{data\_chan}\;
+    \If{a message is available on \VAR{ctrl\_chan}}{%
+      get the message from \VAR{ctrl\_chan}, and push it into \VAR{ctrl\_fifo}\;
+    }
+    \If{a message is available on \VAR{data\_chan}}{%
+      get the message from \VAR{data\_chan}, and push it into \VAR{data\_fifo}\;
+    }
+  }
+\end{algorithm}
+
+\paragraph{Computing thread} The computing thread is in charge of the
+real load management.  As exposed in Algorithm~\ref{algo.comp}, it
+iteratively runs the following operations:
+\begin{itemize}
+\item if some load was received from the neighbors, get it;
+\item if there is some load to send to the neighbors, send it;
+\item run some computation, whose duration is function of the current
+  load of the processor.
+\end{itemize}
+Practically, after the computation, the computing thread waits for a
+small amount of time if the iterations are looping too fast (for
+example, when the current load is near zero).
+
+\begin{algorithm}
+  \caption{Computing thread}
+  \label{algo.comp}
+  \KwData{
+    \begin{algodata}
+      \VAR{data\_fifo} & buffer of received data messages \\
+      \VAR{real\_load} & current load \\
+    \end{algodata}}
+  \While{true}{%
+    \If{\VAR{data\_fifo} is empty and $\VAR{real\_load} = 0$}{%
+      wait until a message is pushed into \VAR{data\_fifo}\;
+    }
+    \While{\VAR{data\_fifo} is not empty}{%
+      pop a message from \VAR{data\_fifo}\;
+      get the load embedded in the message, and add it to \VAR{real\_load}\;
+    }
+    \ForEach{neighbor $n$}{%
+      \If{there is some amount of load $a$ to send to $n$}{%
+        send $a$ units of load to $n$, and subtract it from \VAR{real\_load}\;
+      }
+    }
+    \If{$\VAR{real\_load} > 0.0$}{
+      simulate some computation, whose duration is function of \VAR{real\_load}\;
+      ensure that the main loop does not iterate too fast\;
+    }
+  }
+\end{algorithm}
+
+\paragraph{Load-balancing thread} The load-balancing thread is in
+charge of running the load-balancing algorithm, and exchange the
+control messages.  As shown in Algorithm~\ref{algo.lb}, it iteratively
+runs the following operations:
+\begin{itemize}
+\item get the control messages that were received from the neighbors;
+\item run the load-balancing algorithm;
+\item send control messages to the neighbors, to inform them of the
+  processor's current load, and possibly of virtual load transfers;
+\item wait a minimum (configurable) amount of time, to avoid to
+  iterate too fast.
+\end{itemize}
+
+\begin{algorithm}
+  \caption{Load-balancing}
+  \label{algo.lb}
+  \While{true}{%
+    \While{\VAR{ctrl\_fifo} is not empty}{%
+      pop a message from \VAR{ctrl\_fifo}\;
+      identify the sender of the message,
+      and update the current knowledge of its load\;
+    }
+    run the load-balancing algorithm to make the decision about load transfers\;
+    \ForEach{neighbor $n$}{%
+      send a control messages to $n$\;
+    }
+    ensure that the main loop does not iterate too fast\;
+  }
+\end{algorithm}
+
+\paragraph{}
+For the sake of simplicity, a few details were voluntary omitted from
+these descriptions.  For an exhaustive presentation, we refer to the
+actual source code that was used for the experiments%
+\footnote{As mentioned before, our simulator relies on the SimGrid
+  framework~\cite{casanova+legrand+quinson.2008.simgrid}.  For the
+  experiments, we used a pre-release of SimGrid 3.7 (Git commit
+  67d62fca5bdee96f590c942b50021cdde5ce0c07, available from
+  \url{https://gforge.inria.fr/scm/?group_id=12})}, and which is
+available at
+\url{http://info.iut-bm.univ-fcomte.fr/staff/giersch/software/loba.tar.gz}.
+
+\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?
+par ex, donner l'idée générale de l'implémentation.  l'idée générale est déja décrite en section~\ref{Virtual load}}
+
+\subsection{Experimental contexts}
+\label{Contexts}
+
+In order to assess the performances of our algorithms, we ran our
+simulator with various parameters, and extracted several metrics, that
+we will describe in this section.
+
+\paragraph{Load balancing strategies}
+
+Several load balancing strategies were compared.  We ran the experiments with
+the \emph{Best effort}, and with the \emph{Makhoul} strategies.  \emph{Best
+  effort} was tested with parameter $k = 1$, $k = 2$, and $k = 4$.  Secondly,
+each strategy was run in its two variants: with, and without the management of
+\emph{virtual load}.  Finally, we tested each configuration with \emph{real},
+and with \emph{integer} load.
+
+To summarize the different load balancing strategies, we have:
+\begin{description}
+\item[\textbf{strategies:}] \emph{Makhoul}, or \emph{Best effort} with $k\in
+  \{1,2,4\}$
+\item[\textbf{variants:}] with, or without virtual load
+\item[\textbf{domain:}] real load, or integer load
+\end{description}
+%
+This gives us as many as $4\times 2\times 2 = 16$ different strategies.
+
+\paragraph{End of the simulation}
+
+The simulations were run until the load was nearly balanced among the
+participating nodes.  More precisely the simulation stops when each node holds
+an amount of load at less than 1\% of the load average, during an arbitrary
+number of computing iterations (2000 in our case).
+
+Note that this convergence detection was implemented in a centralized manner.
+This is easy to do within the simulator, but it's obviously not realistic.  In a
+real application we would have chosen a decentralized convergence detection
+algorithm, like the one described by Bahi, Contassot-Vivier, Couturier, and
+Vernier in \cite{10.1109/TPDS.2005.2}.
+
+\paragraph{Platforms}
+
+In order to show the behavior of the different strategies in different
+settings, we simulated the executions on two sorts of platforms.  These two
+sorts of platforms differ by their underlaid network topology.  On the one hand,
+we have homogeneous platforms, modeled as a cluster.  On the other hand, we have
+heterogeneous platforms, modeled as the interconnection of a number of clusters.
+
+The clusters were modeled by a fixed number of computing nodes interconnected
+through a backbone link.  Each computing node has a computing power of
+1~GFlop/s, and is connected to the backbone by a network link whose bandwidth is
+of 125~MB/s, with a latency of 50~$\mu$s.  The backbone has a network bandwidth
+of 2.25~GB/s, with a latency of 500~$\mu$s.
+
+The heterogeneous platform descriptions were created by taking a subset of the
+Grid'5000 infrastructure\footnote{Grid'5000 is a French large scale experimental
+  Grid (see \url{https://www.grid5000.fr/}).}, as described in the platform file
+\texttt{g5k.xml} distributed with SimGrid.  Note that the heterogeneity of the
+platform here only comes from the network topology.  Indeed, since our
+algorithms currently do not handle heterogeneous computing resources, the
+processor speeds were normalized, and we arbitrarily chose to fix them to
+1~GFlop/s.
+
+Then we derived each sort of platform with four different number of computing
+nodes: 16, 64, 256, and 1024 nodes.
+
+\paragraph{Configurations}
+
+The distributed processes of the application were then logically organized along
+three possible topologies: a line, a torus or an hypercube.  We ran tests where
+the total load was initially on an only node (at one end for the line topology),
+and other tests where the load was initially randomly distributed across all the
+participating nodes.  The total amount of load was fixed to a number of load
+units equal to 1000 times the number of node.  The average load is then of 1000
+load units.
+
+For each of the preceding configuration, we finally had to choose the
+computation and communication costs of a load unit.  We chose them, such as to
+have three different computation over communication cost ratios, and hence model
+three different kinds of applications:
+\begin{itemize}
+\item mainly communicating, with a computation/communication cost ratio of $1/10$;
+\item mainly computing, with a computation/communication cost ratio of $10/1$ ;
+\item balanced, with a computation/communication cost ratio of $1/1$.
+\end{itemize}
+
+To summarize the various configurations, we have:
+\begin{description}
+\item[\textbf{platforms:}] homogeneous (cluster), or heterogeneous (subset of
+  Grid'5000)
+\item[\textbf{platform sizes:}] platforms with 16, 64, 256, or 1024 nodes
+\item[\textbf{process topologies:}] line, torus, or hypercube
+\item[\textbf{initial load distribution:}] initially on a only node, or
+  initially randomly distributed over all nodes
+\item[\textbf{computation/communication ratio:}] $10/1$, $1/1$, or $1/10$
+\end{description}
+%
+This gives us as many as $2\times 4\times 3\times 2\times 3 = 144$ different
+configurations.
+%
+Combined with the various load balancing strategies, we had $16\times 144 =
+2304$ distinct settings to evaluate.  In fact, as it will be shown later, we
+didn't run all the strategies, nor all the configurations for the bigger
+platforms with 1024 nodes, since to simulations would have run for a too long
+time.
+
+Anyway, all these the experiments represent more than 240 hours of computing
+time.
+
+\paragraph{Metrics}
+
+In order to evaluate and compare the different load balancing strategies we had
+to define several metrics.  Our goal, when choosing these metrics, was to have
+something tending to a constant value, i.e. to have a measure which is not
+changing anymore once the convergence state is reached.  Moreover, we wanted to
+have some normalized value, in order to be able to compare them across different
+settings.
+
+With these constraints in mind, we defined the following metrics:
+%
+\begin{description}
+\item[\textbf{average idle time:}] that's the total time spent, when the nodes
+  don't hold any share of load, and thus have nothing to compute.  This total
+  time is divided by the number of participating nodes, such as to have a number
+  that can be compared between simulations of different sizes.
+
+  This metric is expected to give an idea of the ability of the strategy to
+  diffuse the load quickly.  A smaller value is better.
+
+\item[\textbf{average convergence date:}] that's the average of the dates when
+  all nodes reached the convergence state.  The dates are measured as a number
+  of (simulated) seconds since the beginning of the simulation.
+
+\item[\textbf{maximum convergence date:}] that's the date when the last node
+  reached the convergence state.
+
+  These two dates give an idea of the time needed by the strategy to reach the
+  equilibrium state.  A smaller value is better.
+
+\item[\textbf{data transfer amount:}] that's the sum of the amount of all data
+  transfers during the simulation.  This sum is then normalized by dividing it
+  by the total amount of data present in the system.
+
+  This metric is expected to give an idea of the efficiency of the strategy in
+  terms of data movements, i.e. its ability to reach the equilibrium with fewer
+  transfers.  Again, a smaller value is better.
+
+\end{description}
+
 
 \subsection{Validation of our approaches}
+\label{Results}
+
+Dans cet ordre:
+...
+- comparer be/makhoul -> be tient la route
+        -> en réel uniquement
+
+- valider l'extension virtual load -> c'est 'achement bien
+- proposer le -k -> ça peut aider dans certains cas
+- conclure avec la version entière -> on n'a pas l'effet d'escalier !
+Q: comment inclure les types/tailles de platesformes ?
+Q: comment faire des moyennes ?
+...
 
+On constate quoi (vérifier avec les chiffres)?
+\begin{itemize}
+\item cluster ou grid, entier ou réel, ne font pas de grosses différences
 
+\item bookkeeping? améliore souvent les choses, parfois au prix d'un retard au démarrage
+
+\item makhoul? se fait battre sur les grosses plateformes
+
+\item taille de plateforme?
+
+\item ratio comp/comm?
+
+\item option $k$? peut-être intéressant sur des plateformes fortement interconnectées (hypercube)
+
+\item volume de comm? souvent, besteffort/plain en fait plus. pourquoi?
+
+\item répartition initiale de la charge ?
+
+\item integer mode sur topo. line n'a jamais fini en plain? vérifier si ce n'est
+  pas à cause de l'effet d'escalier que bk est capable de gommer.
+
+\end{itemize}
+
+\begin{itshape}
 On veut montrer quoi ? :
+\FIXME{remove that part}
 
 1) best plus rapide que les autres (simple, makhoul)
 2) avantage virtual load
@@ -152,7 +682,6 @@ Topologies variées
 Simulation avec temps définies assez long et on mesure la qualité avec : volume de calcul effectué, volume de données échangées
 Mais aussi simulation avec temps court qui montre que seul best converge
 
-
 Expés avec ratio calcul/comm rapide et lent
 
 Quelques expés avec charge initiale aléatoire plutot que sur le premier proc
@@ -166,10 +695,18 @@ On ne tient pas compte de la vitesse des liens donc on la considère homogène
 Prendre un réseau hétérogène et rendre processeur homogène
 
 Taille : 10 100 très gros
+\end{itshape}
 
 \section{Conclusion and perspectives}
 
+\FIXME{conclude!}
+
+\begin{acknowledgements}
+  Computations have been performed on the supercomputer facilities of
+  the Mésocentre de calcul de Franche-Comté.
+\end{acknowledgements}
 
+\FIXME{find and add more references}
 \bibliographystyle{spmpsci}
 \bibliography{biblio}
 
@@ -178,8 +715,11 @@ Taille : 10 100 très gros
 %%% Local Variables:
 %%% mode: latex
 %%% TeX-master: t
+%%% fill-column: 80
 %%% 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 isend irecv Cortés et al chan ctrl fifo Makhoul GFlop xml pre
+% LocalWords:  FEMTO Makhoul's fca bdee cdde Contassot Vivier underlaid