From: couchot Date: Thu, 28 Nov 2013 20:25:52 +0000 (+0100) Subject: corrections typos v0 X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/commitdiff_plain/465f4bf04069c64e49bc9420f434a0b96cbcd127 corrections typos v0 --- diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex index 456b369..183e00b 100644 --- a/IWCMC14/HLG.tex +++ b/IWCMC14/HLG.tex @@ -1,52 +1,41 @@ -Let us first the basic recalls of the~\cite{HLG09} article. - - -The precise the context of video sensor network as represented for instance -in figure~\ref{fig:sn}. - \begin{figure} \begin{center} -\includegraphics[scale=0.5]{reseau.png} +\includegraphics[scale=0.3]{reseau.png} + +\begin{scriptsize} +An example of a sensor network ofsize 10. All nodes are video sensor +except the 5 and the 9 one which is the sink. +\JFC{reprendre la figure, trouver un autre titre} +\end{scriptsize} + \caption{SN with 10 sensor}\label{fig:sn}. \end{center} \end{figure} - -Let us give a formalisation of such a video network sensor. -We start with the flow formalising: - -The video sensor network is represented as a strongly -connected oriented labelled graph. +Let us first recall the basics of the~\cite{HLG09} article. +The video sensor network is memorized as a connected non oriented +oriented labelled graph. In this one, -the nodes, in a set $N$ are sensors, links, or the sink. +the nodes, in a set $N$, are sensors, links, or the sink. Furthermore, there is an edge from $i$ to $j$ if $i$ can send a message to $j$. The set of all edges is further denoted as -$L$ . -This boolean information is stored as a +$L$. +Figure~\ref{fig:sn} gives an example of such a network. + +This link information is stored as a matrix $A=(a_{il})_{i \in N, l \in L}$, where -$a_{il} = -\left\{ - \begin{array}{rl} - 1 & \textrm{if $l$ starts with $i$ } \\ - -1 & \textrm{si $l$ ends width $i$ } \\ - 0 & \textrm{otherwise} - \end{array} - \right.$. +$a_{il}$ is 1 if $l$ starts with $i$, is -1 if $l$ ends width $i$ +and 0 otherwise. Let $V \subset N $ be the set of the video sensors of $N$. -Let thus $R_h$, $R_h \geq 0$ be the encoding rate of video sensor $h$, $h \in V$. -Let $\eta_{hi}$ be the production rate of the $i$ node, for the $h$ session. More precisely, we have - $$ -\eta_{hi} = -\left\{ - \begin{array}{rl} - R_h & \textrm{if $i$ is $h$} \\ - -R_h & \textrm{if $i$ is the sink} \\ - 0 & \textrm{otherwise} - \end{array} - \right.$$ +Let thus $R_h$, $R_h \geq 0$, +be the encoding rate of video sensor $h$, $h \in V$. +Let $\eta_{hi}$ be the production rate of the node $i$, +for the session initiated by $h$. More precisely, we have +$ \eta_{hi}$ is equal to $ R_h$ if $i$ is $h$, +is equal to $-R_h$ if $i$ is the sink, and $0$ otherwise. We are then left to focus on the flows in this network. Let $x_{hl}$, $x_{hl}\geq 0$, be the flow inside the edge $l$ that @@ -82,14 +71,18 @@ The objective is thus to find $R$, $x$, $P_s$ which minimize \end{enumerate} -To achieve a local optimisation, the problem is translated into an +To achieve this optimizing goal +a local optimisation, the problem is translated into an equivalent one: find $R$, $x$, $P_s$ which minimize $\sum_{i \in N }q_i^2$ with the same set of constraints, but item \ref{itm:q}, which is replaced by: - -$$P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) + -\sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q.B_i, \forall i \in N$$ - +$$ +\begin{array}{l} +P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) \\ +\qquad + + \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q.B_i, \forall i \in N +\end{array} +$$ The authors then apply a dual based approach with Lagrange multiplier to solve such a problem. @@ -114,11 +107,11 @@ The proposed algorithm iteratively computes the following variables \item $v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} - \theta^{(k)}.\left( R_h^{(k)} - \dfrac{\ln(\sigma^2/D_h)}{\gamma.(P_{sh}^{(k)})^{2/3}} \right)\right\}$ \item - $\begin{array}{rcl} - \lambda_{i}^{(k+1)} = \lambda_{i}^{(k)} - \theta^{(k)}&.&\left( - q^{(k)}.B_i - - \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl}^{(k)} \right) \right. \\ - && - \left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)} \right) + $\begin{array}{l} + \lambda_{i}^{(k+1)} = \lambda_{i}^{(k)} - \theta^{(k)}.\left( + q^{(k)}.B_i \right.\\ + \qquad\qquad\qquad -\sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl}^{(k)} \right) \\ + \qquad\qquad\qquad - \left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)} \right) \end{array} $ @@ -162,16 +155,20 @@ $ \item $ x_{hl}^{(k)} = +\begin{array}{l} \arg \min_{x \geq 0} \left( -\delta.x^2 + x. +\delta.x^2 \right.\\ +\qquad \qquad + x. \sum_{i \in N} \left( \lambda_{i}^{(k)}.(c^s_l.a_{il}^{+} + -c^r. a_{il}^{-} )+ - u_{hi}^{(k)} a_{il} +c^r. a_{il}^{-} ) \right.\\ +\qquad \qquad\qquad \qquad + +\left.\left. u_{hi}^{(k)} a_{il} \right) \right) - $ +\end{array} +$ \end{enumerate} where the first four elements are dual variable and the last four ones are primal ones \ No newline at end of file diff --git a/IWCMC14/main.tex b/IWCMC14/main.tex index f77d4eb..0bea088 100644 --- a/IWCMC14/main.tex +++ b/IWCMC14/main.tex @@ -1,4 +1,4 @@ -\documentclass{IEEEtran} +\documentclass[10pt]{IEEEtran} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[english]{babel}