Normalize labels.

[loba-papers.git] / loba-besteffort / loba-besteffort.tex

diff --git a/loba-besteffort/loba-besteffort.tex b/loba-besteffort/loba-besteffort.tex
index b53ffec9d55c835ce200c1710d3266c37f11200f..d774cce15fce8c3c4ad4df8dcc98c3458b92fa4f 100644 (file)
--- a/loba-besteffort/loba-besteffort.tex
+++ b/loba-besteffort/loba-besteffort.tex
@@ -3,9 +3,9 @@
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 

-\usepackage{newtxtext}
-\usepackage[cmintegrals]{newtxmath}
-%\usepackage{mathptmx,helvet,courier}
+%\usepackage{newtxtext}
+%\usepackage[cmintegrals]{newtxmath}
+\usepackage{mathptmx,helvet,courier}

 
 \usepackage{amsmath}
 \usepackage{graphicx}
 
 \usepackage{amsmath}
 \usepackage{graphicx}


@@ -134,20 +134,21 @@ 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.
 
 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.  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.
+In the following of this paper, Section~\ref{sec.bt-algo} describes the
+Bertsekas and Tsitsiklis' asynchronous load balancing algorithm. Moreover, we
+present a possible problem in the convergence conditions.
+Section~\ref{sec.besteffort} 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{sec.other}.  In
+Section~\ref{sec.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{sec.simulations}.  Finally we give a
+conclusion and some perspectives to this work.

 
 
 
 \section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}
 
 
 
 \section{Bertsekas  and Tsitsiklis' asynchronous load balancing algorithm}

-\label{BT algo}
+\label{sec.bt-algo}

 
 In  order  prove  the  convergence  of  asynchronous  iterative  load  balancing
 Bertsekas         and        Tsitsiklis         proposed         a        model
 
 In  order  prove  the  convergence  of  asynchronous  iterative  load  balancing
 Bertsekas         and        Tsitsiklis         proposed         a        model


@@ -170,7 +171,7 @@ 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)
 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}
+\label{eq.ping-pong}

 \end{equation}
 
 
 \end{equation}
 
 


@@ -193,9 +194,9 @@ x_2(t)=100   \\
 x_3(t)=99.99\\
  x_3^2(t)=99.99\\
 \end{eqnarray*}
 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
+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.
 $x_3^2(t)$.  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.


@@ -209,7 +210,7 @@ that they are sufficient to ensure the convergence of the load-balancing
 algorithm.
 
 \section{Best effort strategy}
 algorithm.
 
 \section{Best effort strategy}

-\label{Best-effort}
+\label{sec.besteffort}

 
 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,
 
 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,


@@ -281,12 +282,12 @@ 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
 
 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~\ref{sec.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{sec.results}]{}

 
 \section{Other strategies}
 
 \section{Other strategies}

-\label{Other}
+\label{sec.other}

 
 Another load balancing strategy, working under the same conditions, was
 previously developed by Bahi, Giersch, and Makhoul in
 
 Another load balancing strategy, working under the same conditions, was
 previously developed by Bahi, Giersch, and Makhoul in


@@ -307,7 +308,7 @@ neighbor.
 
 
 \section{Virtual load}
 
 
 \section{Virtual load}

-\label{Virtual load}
+\label{sec.virtual-load}

 
 In this section,  we present the concept of \emph{virtual load}.  In order to
 use this concept, load balancing messages must be sent using two different kinds
 
 In this section,  we present the concept of \emph{virtual load}.  In order to
 use this concept, load balancing messages must be sent using two different kinds


@@ -337,7 +338,7 @@ information of the load they will receive, so they can take in into account.
 \FIXME{describe integer mode}
 
 \section{Simulations}
 \FIXME{describe integer mode}
 
 \section{Simulations}

-\label{Simulations}
+\label{sec.simulations}

 
 In order to test and validate our approaches, we wrote a simulator
 using the SimGrid
 
 In order to test and validate our approaches, we wrote a simulator
 using the SimGrid


@@ -348,13 +349,12 @@ 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.
 
 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}.
+The simulation model is detailed in the next section (\ref{sec.model}), and the
+experimental contexts are described in section~\ref{sec.exp-context}.  Then the
+results of the simulations are presented in section~\ref{sec.results}.

 
 \subsection{Simulation model}
 
 \subsection{Simulation model}

-\label{Sim model}
+\label{sec.model}

 
 In the simulation model the processors exchange messages which are of
 two kinds.  First, there are \emph{control messages} which only carry
 
 In the simulation model the processors exchange messages which are of
 two kinds.  First, there are \emph{control messages} which only carry


@@ -491,10 +491,11 @@ iteratively runs the following operations:
 \end{algorithm}
 
 \paragraph{}\FIXME{ajouter des détails sur la gestion de la charge virtuelle ?
 \end{algorithm}
 
 \paragraph{}\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}}
+  par ex, donner l'idée générale de l'implémentation.  l'idée générale est déja
+  décrite en section~\ref{sec.virtual-load}}

 
 \subsection{Experimental contexts}
 
 \subsection{Experimental contexts}

-\label{Contexts}
+\label{sec.exp-context}

 
 In order to assess the performances of our algorithms, we ran our
 simulator with various parameters, and extracted several metrics, that
 
 In order to assess the performances of our algorithms, we ran our
 simulator with various parameters, and extracted several metrics, that


@@ -643,7 +644,7 @@ With these constraints in mind, we defined the following metrics:
 
 
 \subsection{Experimental results}
 
 
 \subsection{Experimental results}

-\label{Results}
+\label{sec.results}

 
 In this section, the results for the different simulations will be presented,
 and we'll try to explain our observations.
 
 In this section, the results for the different simulations will be presented,
 and we'll try to explain our observations.