quelques modifs

[desynchronisation-controle.git] / IWCMC14 / HLG.tex

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index ff12b04fa9d612566bf61cd45a0eb129bcb9809d..998f94b5aca6f97d90777f699b241e382fd812f8 100644 (file)
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -1,24 +1,30 @@
-\begin{figure}
+\begin{figure*}

 \begin{center}
 \begin{center}

-\includegraphics[scale=0.3]{reseau.png}

 
 

+\includegraphics[scale=0.2]{SensorNetwork.png}

 \begin{scriptsize}
 \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}.
+An example of a sensor network of size 10. 
+All nodes are video sensors (depicted as small discs)
+except the 9 one which is the sink (depicted as a rectangle). 
+Large lircles represent the maximum 
+transmission range which is set to 20 in a square region which is 
+$50 m \times  50 m$.
+\end{scriptsize} 
+\caption{Illustration of a Sensor Network of size 10}\label{fig:sn}.

 \end{center}
 \end{center}

-\end{figure} 
+\end{figure*} 

 
 Let us first recall  the basics of the~\cite{HLG09} article.
 The video sensor network is memorized as a connected non oriented 
 
 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. 
+graph. 

 In this one, 
 the nodes, in a set $N$, are sensors, links, or the sink.
 Furthermore, there is an edge from $i$ to $j$ if $i$ can 
 In this one, 
 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
+send a message to $j$, \textit{i. e.}, the distance betwween 
+$i$ and $j$ is less than a given maximum 
+transmission range.
+All the possible edges are stored into a sequence  

 $L$.
 Figure~\ref{fig:sn} gives an example of such a network.
   
 $L$.
 Figure~\ref{fig:sn} gives an example of such a network.
   


@@ -27,44 +33,65 @@ matrix $A=(a_{il})_{i \in N, l \in L}$,
 where 
 $a_{il}$ is  1 if $l$ starts with $i$, is -1 if  $l$ ends width $i$ 
 and  0 otherwise.
 where 
 $a_{il}$ is  1 if $l$ starts with $i$, is -1 if  $l$ ends width $i$ 
 and  0 otherwise.

-
+Moreover, the outgoing  links(resp. the incoming links) are represented 
+with the $A^+$ matrix (res. with the $A^-$ matrix), whose elements are defined:
+$a_{il}^+$ (resp. $a_{il}^-$) is 1  if the link $l$ is an outgoing link from $i$ 
+(resp an incoming link into $i$) 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 $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  node $i$, 
-for the  session initiated by $h$. More precisely, we have 
+Let $\eta_{hi}$ be the  rate inside the  node $i$ 
+of the production that has beeninitiated 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.
   
 $ \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 us focus on the flows in this network.

 Let $x_{hl}$, $x_{hl}\geq 0$, be the flow inside the edge $l$ that 
 Let $x_{hl}$, $x_{hl}\geq 0$, be the flow inside the edge $l$ that 

-issued from the $h$ session and 
+issued from the node $h$  and 

 let $y_l = \sum_{h \in V}x_{hl} $ the sum of all the flows inside $l$.
 Thus, what is produced inside the $i^{th}$ sensor for session $h$ 
 is  $ \eta_{hi} = \sum_{l \in L }a_{il}x_{hl} $.
 
 
 The encoding power of the $i$ node is $P_{si}$, $P_{si} > 0$.
 let $y_l = \sum_{h \in V}x_{hl} $ the sum of all the flows inside $l$.
 Thus, what is produced inside the $i^{th}$ sensor for session $h$ 
 is  $ \eta_{hi} = \sum_{l \in L }a_{il}x_{hl} $.
 
 
 The encoding power of the $i$ node is $P_{si}$, $P_{si} > 0$.

-
-The distortion is bounded $\sigma^2 e^{-\gamma . R_h.P_{sh}^{}2/3} \leq D_h$.
-
+The emmission distortion  of the $i$ node is 
+$\sigma^2 e^{-\gamma . R_i.P_{si}^{}2/3}$
+where $\sigma^2$ is the average input variance and
+$\gamma$ is the encoding efficiency coefficient.
+This distortion  
+is bounded by a constant value $D_h$.

 The initial energy of the $i$ node is  $B_i$.
 The initial energy of the $i$ node is  $B_i$.

-
+The transmission consumed power of node $i$ is  
+$P_{ti} = c_l^s.y_l$ where  $c_l^s$ is the transmission energy
+consumption cost of link $l$, $l\in L$. This cost is defined 
+as foolows:  $c_l^s = \alpha +\beta.d_l^{n_p} $ where 
+$d_l$ represents the distance of the link $l$,
+$\alpha$, $\beta$, and $n_p$ are constant. 
+The reception consumed power of node $i$ is  
+$P_{ri} = c^r \sum_{l \in L } a_{il}^-.y_l$ 
+where  $c^r$ is a reception energy consumption cost.

 The overall consumed power of the $i$ node is 
 $P_{si}+ P_{ti} + P_{ri}= 
 P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l + 
 The overall consumed power of the $i$ node is 
 $P_{si}+ P_{ti} + P_{ri}= 
 P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l + 

-\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i. 
-$
-
-The objective is thus to find $R$, $x$, $P_s$  which minimize
- $q$ under the following set of constraints
+\sum_{l \in L} a_{il}^{-}.c^r.y_l $.
+%\leq q.B_i. 
+%$
+
+The objective is thus to find $R$, $x$, $P_s$  which maximizes
+the network lifetime $T_{\textit{net}}$, or equivalently which minimizes
+$q=1/{T_{\textit{net}}}$. 
+Let $B_i$ is the initial energy in node $i$.
+One have the equivalent objective to find $R$, $x$, $P_s$ which minimizes
+$q^2$
+under the following set of constraints:

 \begin{enumerate}
 \item $\sum_{l \in L }a_{il}x_{hl} = \eta_{hi},\forall h \in V, \forall i \in N  $
 \item $ \sum_{h \in V}x_{hl} = y_l,\forall l \in L$
 \item $\dfrac{\ln(\sigma^2/D_h)}{\gamma.P_{sh}^{2/3}} \leq R_h \forall h \in V$
 \item \label{itm:q} $P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l + 
 \begin{enumerate}
 \item $\sum_{l \in L }a_{il}x_{hl} = \eta_{hi},\forall h \in V, \forall i \in N  $
 \item $ \sum_{h \in V}x_{hl} = y_l,\forall l \in L$
 \item $\dfrac{\ln(\sigma^2/D_h)}{\gamma.P_{sh}^{2/3}} \leq R_h \forall h \in V$
 \item \label{itm:q} $P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l + 

-\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i, \forall i \in N$
+c^r.\sum_{l \in L} a_{il}^{-}.y_l \leq q.B_i, \forall i \in N$
+\item $\sum_{i \in N} a_{il}q_i = 0, \forall l \in L$ 

 \item $x_{hl}\geq0, \forall h \in V, \forall l \in L$
 \item $R_h \geq 0, \forall h \in V$
 \item $P_{sh} > 0,\forall h \in V$
 \item $x_{hl}\geq0, \forall h \in V, \forall l \in L$
 \item $R_h \geq 0, \forall h \in V$
 \item $P_{sh} > 0,\forall h \in V$


@@ -79,9 +106,12 @@ $$
 \begin{array}{l}
 P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) \\
 \qquad + 
 \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
+ \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q_i.B_i, \forall i \in N

 \end{array}
 $$
 \end{array}
 $$

+and where the following constraint is added
+$$ $q_i > 0, \forall i \in N  $$
+

 
 
 They thus replace the objective of reducing
 
 
 They thus replace the objective of reducing


@@ -94,7 +124,7 @@ by the objective of reducing
 \label{eq:obj2}
 \end{equation}
 where $\delta$ is a regularisation factor.
 \label{eq:obj2}
 \end{equation}
 where $\delta$ is a regularisation factor.

-This indeed introduces quadratic fonctions on variables $x_{hl}$ and 
+This indeed introduces quadratic functions on variables $x_{hl}$ and 

 $R_{h}$ and makes some of the functions strictly convex.
 
 The authors then apply a classical dual based approach with Lagrange multiplier 
 $R_{h}$ and makes some of the functions strictly convex.
 
 The authors then apply a classical dual based approach with Lagrange multiplier 


@@ -130,7 +160,7 @@ c^r. a_{il}^{-} ) \right.\\
 \end{equation}
 
 The proposed algorithm iteratively computes the following variables 
 \end{equation}
 
 The proposed algorithm iteratively computes the following variables 

-untill the variation of the dual function is less than a given threshold.
+until the variation of the dual function is less than a given threshold.

 \begin{enumerate}
 \item $ u_{hi}^{(k+1)} = u_{hi}^{(k)} - \theta^{(k)}. \left(
  \eta_{hi}^{(k)} - \sum_{l \in L }a_{il}x_{hl}^{(k)} \right) $
 \begin{enumerate}
 \item $ u_{hi}^{(k+1)} = u_{hi}^{(k)} - \theta^{(k)}. \left(
  \eta_{hi}^{(k)} - \sum_{l \in L }a_{il}x_{hl}^{(k)} \right) $


@@ -139,7 +169,7 @@ $v_{h}^{(k+1)}= \max\left\{0,v_{h}^{(k)} -  \theta^{(k)}.\left( R_h^{(k)} - \dfr
 \item 
   $\begin{array}{l}
    \lambda_{i}^{(k+1)} = \max\left\{0, \lambda_{i}^{(k)} - \theta^{(k)}.\left( 
 \item 
   $\begin{array}{l}
    \lambda_{i}^{(k+1)} = \max\left\{0, \lambda_{i}^{(k)} - \theta^{(k)}.\left( 

-    q^{(k)}.B_i \right. \left.\\
+    q^{(k)}.B_i \right. \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.\left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)}  \right) \right\}
 \end{array}
   \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.\left. \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl}^{(k)} \right) - P_{si}^{(k)}  \right) \right\}
 \end{array}


@@ -168,7 +198,7 @@ P_{sh}^{(k)}
 =
 \arg \min_{p > 0} 
 \left(
 =
 \arg \min_{p > 0} 
 \left(

-v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p ^{2/3}} + \lambda_h^{(k)}p
+v_h^{(k)}.\dfrac{\ln(\sigma^2/D_h)}{\gamma p^{2/3}} + \lambda_h^{(k)}p

 \right)
 $
 
 \right)
 $