From c4e399c461dd18f8ebd98333cc45aaf32a6769a4 Mon Sep 17 00:00:00 2001
From: couchot <jf.couchot@gmail.com>
Date: Wed, 4 Dec 2013 21:20:07 +0100
Subject: [PATCH] quelques corrections en +

---
 IWCMC14/HLG.tex | 28 +++++++++++++++++++---------
 1 file changed, 19 insertions(+), 9 deletions(-)

diff --git a/IWCMC14/HLG.tex b/IWCMC14/HLG.tex
index 7daf581..998f94b 100644
--- a/IWCMC14/HLG.tex
+++ b/IWCMC14/HLG.tex
@@ -11,7 +11,7 @@ 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 SN of size 10}\label{fig:sn}.
+\caption{Illustration of a Sensor Network of size 10}\label{fig:sn}.
 \end{center}
 \end{figure*} 
 
@@ -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 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  
@@ -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
-- 
2.39.5