## 5 Cooperative Scheduling for Object Detection in Cluster-Based WMSN.
### Power Conservation Evaluation  
To evaluate the proposed scheduling scheme in terms of power conservation, we compare the cooperative scheduled scheme with a single-tier network or a tier of a multi-tier architecture consisting of N nodes performing object detection without coordination among them as [14,15,18], in which, nodes are awakened with a time period of T. We note that the evaluation is over the sensing subsystem and that the radio subsystem (_i.e._, transmission and reception of packets) is not taken into account. The energy consumed in the network for object detection by N nodes during a duty-cycle interval of T in the non-collaborative scheduling is:  
\[\mathrm{E}=\mathrm{N}\cdot(\mathrm{T}_{\mathrm{sleep}}\cdot\mathrm{P}_{ \mathrm{sleep}}+\mathrm{E}_{\mathrm{w}\_\mathrm{app}}+\mathrm{E}_{\mathrm{cap} }+\mathrm{E}_{\mathrm{detect}}) \tag{14}\]  
where \(\mathrm{T}_{\mathrm{sleep}}\) and \(\mathrm{P}_{\mathrm{sleep}}\) are the period and power consumption for a node in sleep mode. \(\mathrm{E}_{\mathrm{w}\_\mathrm{app}}\), \(\mathrm{E}_{\mathrm{cap}}\) and \(\mathrm{E}_{\mathrm{detect}}\) respectively are the energies consumed in waking up a node, capturing a picture and performing object detection. Let us now consider the cooperative scheduling algorithm in a clustered tier/network. Both, the interval between waking up consecutive nodes in the same cluster and the period of waking up a given node are functions of the cluster-size of the cluster which the nodes belong to. In one hand, in clusters with high cluster-size, \(\mathrm{T}_{\mathrm{interval}}\) is small and thus cluster duty-cycle frequency is increased. On the other hand, higher number of nodes in the cluster causes longer periods \(\mathrm{T}_{\mathrm{P}}\) for awaking a given node of the cluster and thus yields an enhancement for power conservation in cluster's members. Assuming average cluster-size for all clusters in the tier/network, \(\mathrm{T}_{\mathrm{P}}\) will be:  
Figure 10: _Cont._\[\mathrm{T_{p}}=\frac{\mathrm{T}\cdot\mu_{\mathrm{C_{size}}}}{\mu_{\mathrm{C_{size}}} -\gamma\cdot(\mu_{\mathrm{C_{size}}}-1)} \tag{15}\]  
where \(\mathrm{T}\) is the base period for waking nodes in the base un-coordinated tier. Figure 11 shows the evolution of \(\mathrm{T_{p}}\) normalized by \(\mathrm{T}\) (_i.e._, \(\mu_{\mathrm{Csize}}/\beta\)) for several node densities and overlapping scales, \(\gamma\). We may observe that the node average duty-cycle frequency is reduced by factors that are, for example, on the order of 0.78 for a 200 node network and a scale factor of \(\gamma=0.6\). **Figure 11.** The average of coefficient \(\mu_{\mathrm{Csize}}/\beta\) for several node densities and overlapping scales, \(\gamma\). Consequently, the total amount of averaged consumed energy by nodes for object detection in the coordinated tier during \(\mathrm{T_{p}}\) will be:  
\[\mathrm{E_{p}}=\mathrm{E}+\mathrm{N}\cdot\mathrm{P_{sleep}}\cdot(\mathrm{T_{p}}- \mathrm{T}) \tag{16}\]  
From Equations (15) and (16) we have:  
\[E_{p}=E+\frac{\gamma\cdot\mathrm{T}\cdot(\mu_{\mathrm{C_{size}}}-1)}{\mu_{ \mathrm{C_{size}}}-\gamma\cdot(\mu_{\mathrm{C_{size}}}-1)}\cdot\mathrm{N} \cdot\mathrm{P_{sleep}}\]  
So:  
\[\frac{\mathrm{E_{p}}}{\mathrm{T_{p}}}=(1-\frac{\mu_{\mathrm{C_{size}}}-1}{ \mu_{\mathrm{C_{size}}}}\cdot\gamma)\cdot\frac{\mathrm{E}}{\mathrm{T}}+\frac {\mathrm{N}\cdot\gamma\cdot(\mu_{\mathrm{C_{size}}}-1)}{\mu_{\mathrm{C_{size} }}}\cdot\mathrm{P_{sleep}}\ \ \ \mathrm{where}\ \ (0<\gamma<1)\ \ \mathrm{and}\ \ (\mu_{\mathrm{C_{size}}}>1)\]Therefore, the consumed power is:  
\[\mathrm{P_{p}}=\lambda\cdot\mathrm{P}+\sigma\cdot\mathrm{P_{sleep}} \tag{17}\]  
where:  
\[\lambda=(1-\frac{\mu_{\mathrm{C_{size}}}-1}{\mu_{\mathrm{C_{size}}}}\cdot\gamma) \quad\quad,\ 0<\lambda<1\]  
\[\sigma=\frac{\mathrm{N}\cdot\gamma\cdot(\mu_{\mathrm{C_{size}}}-1)}{\mu_{ \mathrm{C_{size}}}}\quad\quad,\ 0<\sigma<\gamma\cdot\mathrm{N}\]  
Parameter P in Equation (17) is the power consumed in the network with the base un-coordinated mechanism.