X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/17a131a2d00611e1582103549a9c7d2bac2be03e..74148119c4eab3cb701943974e740e25f9ab0c7a:/IWCMC14/HLG.tex?ds=sidebyside

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index 7daf581..1f9fab9 100644
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -7,11 +7,11 @@
 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 
+Large circles 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 SN of size 10}\label{fig:sn}.
+\caption{Illustration of a Sensor Network of size 10}\label{fig:sn}.
 \end{center}
 \end{figure*} 
 
@@ -21,7 +21,7 @@ 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 
-send a message to $j$, \textit{i. e.}, the distance betwween 
+send a message to $j$, \textit{i. e.}, the distance between 
 $i$ and $j$ is less than a given maximum 
 transmission range.
 All the possible edges are stored into a sequence  
@@ -42,7 +42,7 @@ 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  rate inside the  node $i$ 
-of the production that has beeninitiated by $h$. More precisely, we have 
+of the production that has been 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.
   
@@ -55,7 +55,7 @@ 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 emmission distortion  of the $i$ node is 
+The emission 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.
@@ -65,7 +65,7 @@ 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} $ wehre 
+as follows:  $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  
@@ -74,17 +74,24 @@ where  $c^r$ is a reception energy consumption cost.
 The overall consumed power of the $i$ node is 
 $P_{si}+ P_{ti} + P_{ri}= 
 P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l + 
-\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i. 
-$
-
-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} $P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.y_l + 
-\sum_{l \in L} a_{il}^{-}.c^r.y_l \leq q.B_i, \forall i \in N$
+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 $P_{sh} > 0,\forall h \in V$
@@ -99,9 +106,12 @@ $$
 \begin{array}{l}
 P_{si}+ \sum_{l \in L}a_{il}^{+}.c^s_l.\left( \sum_{h \in V}x_{hl} \right) \\
 \qquad + 
- \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q.B_i, \forall i \in N
+ \sum_{l \in L} a_{il}^{-}.c^r.\left( \sum_{h \in V}x_{hl} \right) \leq q_i.B_i, \forall i \in N
 \end{array}
 $$
+and where the following constraint is added
+$q_i > 0, \forall i \in N$.
+
 
 
 They thus replace the objective of reducing
@@ -118,20 +128,20 @@ 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 
-to solve such a problem~\cite{}.
+to solve such a problem~\cite{PM06}.
 They first introduce dual variables 
 $u_{hi}$, $v_{h}$, $\lambda_{i}$, and  $w_l$ for any 
-$h \in V$,$ i \in N$, and $l \in L$.
+$h \in V$, $ i \in N$, and $l \in L$.
 
 \begin{equation}
 \begin{array}{l}
 L(R,x,P_{s},q,u,v,\lambda,w)=\\ 
-\sum_{i \in N} q_i^2 + q_i.  \left(
-\sum_{l \in L } a_{il}w_l^{(k)}-
+\sum_{i \in N} \left( q_i^2 + q_i.  \left(
+\sum_{l \in L } a_{il}w_l-
 \lambda_iB_i
-\right)\\
-+ \sum_{h \in V}
-v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma P_{sh} ^{2/3}} + \lambda_h P_{sh}\\
+\right)\right) \\
++ \sum_{h \in V} \left(
+v_h.\dfrac{\ln(\sigma^2/D_h)}{\gamma P_{sh} ^{2/3}} + \lambda_h P_{sh} \right)\\
 + \sum_{h \in V} \sum_{l\in L}
 \left(
 \delta.x_{hl}^2  \right.\\
@@ -143,10 +153,11 @@ c^r. a_{il}^{-} ) \right.\\
 \left.\left. u_{hi} a_{il}
 \right)
 \right)\\
- + \sum_{h \in V}
+ + \sum_{h \in V} \left(
 \delta R_{h}^2 
--v_h.R_{h} - \sum_{i \in N} u_{hi}\eta_{hi}
+-v_h.R_{h} - \sum_{i \in N} u_{hi}\eta_{hi}\right)
 \end{array}
+\label{eq:dualFunction}
 \end{equation}
 
 The proposed algorithm iteratively computes the following variables 
@@ -159,21 +170,21 @@ $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( 
-    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\}
+    q^{(k)}.B_i - P_{si}^{(k)} \right. \right.\\
+  \qquad\qquad\qquad -\sum_{l \in L}a_{il}^{+}.c^s_l. \sum_{h \in V}x_{hl}^{(k)}    \\
+  \qquad\qquad\qquad  - \left.\left. c^r.\sum_{l \in L} a_{il}^{-}. \sum_{h \in V}x_{hl}^{(k)} \right)   \right\}
 \end{array}
 $
 
 \item 
-$w_l^{(k+1)} = w_l^{(k+1)} +  \theta^{(k)}. \left( \sum_{i \in N} a_{il}.q_i^{(k)} \right)$
+$w_l^{(k+1)} = w_l^{(k+1)} +  \theta^{(k)}.  \sum_{i \in N} a_{il}.q_i^{(k)} $
 
 
 \item 
-$\theta^{(k)} = \omega / t^{1/2}$
+$\theta^{(k)} = \omega / k^{1/2}$
 
  \item 
-$q_i^{(k)} = \arg\min_{q_i>0}
+$q_i^{(k+1)} = \arg\min_{q_i>0}
 \left(
 q_i^2 + q_i. 
 \left(
@@ -184,7 +195,7 @@ q_i^2 + q_i.
 
 \item \label{item:psh} 
 $
-P_{sh}^{(k)} 
+P_{sh}^{(k+1)} 
 =
 \arg \min_{p > 0} 
 \left(
@@ -194,7 +205,7 @@ $
 
 \item 
 $
-R_h^{(k)}
+R_h^{(k+1)}
 =
 \arg \min_{r \geq 0 }
 \left(
@@ -204,7 +215,7 @@ R_h^{(k)}
 $
 \item 
 $
-x_{hl}^{(k)} =
+x_{hl}^{(k+1)} =
 \begin{array}{l}
 \arg \min_{x \geq 0}
 \left(
@@ -220,5 +231,4 @@ c^r. a_{il}^{-} ) \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
+for any $h \in V$, $i \in N$, and $l \in L$. 
