X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/desynchronisation-controle.git/blobdiff_plain/17a131a2d00611e1582103549a9c7d2bac2be03e..refs/heads/master:/IWCMC14/HLG.tex?ds=inline 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 $l \in L$.