## 5 Cooperative Scheduling for Object Detection in Cluster-Based WMSN.
### Power Conservation Evaluation
The consumed power in our scheme (\(\mathrm{P_{P}}\)) is reduced by a factor \(\lambda\) with respect to P. The \(\lambda\) factor depends on the average cluster-size and the overlapping scale factor. As can be observed from Equation (17) increasing \(\mu_{\mathrm{Csize}}\) produces lower values of \(\lambda\), and thus a saving in energy with respect the uncoordinated system. For example a \(\mu_{\mathrm{Csize}}=1.5\) (100 nodes with \(\gamma=0.5\)) produces a \(\lambda=1-\gamma/3=0.83\) while a \(\mu_{\mathrm{Csize}}=2.15\) (200 nodes with \(\gamma=0.5\)) produces a \(\lambda=1-0.53\)\(\gamma=0.73\). The other term (\(\sigma\cdot\mathrm{P_{sleep}}\)) in Equation (17) is due to the fact of taking nodes to sleep mode in intervals of duration (\(\mathrm{T_{P}>T}\)) and then nodes sleep \(\mathrm{T_{P}}\)-T more time than in the un-clustered scheme. Figure 12 illustrates the impact of factor \(\lambda\) in Equation (17) in terms of node densities for several overlapping scales. From this figure we can see that in high node density tiers, the factor \(\lambda\) is more beneficial since \(\mu_{\mathrm{Csize}}\) is higher and thus there is more potential of cooperation among nodes. Figure 13 shows the consumed power (P) in the base un-coordinated tier for object detection in four cases of period of duty-cycle for different node densities. The consumed power has been computed for  
Figure 10:0 Factor \(\lambda\) in cooperative scheduling for several overlapping scales. nodes consisting of Cyclops as camera sensor embedded in the host MICA II, similar to the tier 1 in [15]. For instance, in the case without coordination, the power consumed in a tier consisting of 200 nodes that performs object detection with a duty cycle of T = 5 s, is 1.344 watts. In the coordinated network with the same number of nodes and an overlapping scale of 0.5, the power consumed by the network would be reduced by a factor \(\lambda\) of 0.737 (see Figure 12) at the cost of increasing 52.60 mW, (\(\sigma\cdot\)P\({}_{\text{sleep}}\)). This means a tier power consumption of 1.344 \(\times\)0.737 + 0.0526 = 1.043 Watts implying a reduction of 22.39%. Thus, in this case, the Prolongation Lifetime Ratio (PLR) would be of 1.344/1.043 = 1.289. Figure 14a,b shows the prolongation lifetime ratio assuming an overlapping scale of 0.5 and 0.6 for different node densities in four cases of duty-cycle (T). Tiers with high number of nodes have higher capability for cooperation and thus their nodes can conserve considerable amount of energy comparing to sparse networks and consequently, have longer prolonged lifetime. The figure indicates the more prolongation lifetime for dense tiers. Finally, the lifetime increment in a deployment of 250 nodes is shown in Figure 10:1 The lifetime is normalized to L\({}_{\text{t}}\), the time to which all nodes would deplete their batteries without coordination. This lifetime only is due to sensing and thus forwarding and other tasks are not taken into account. We assume that all nodes under an uncoordinated duty-cycle scheme will die at the same time with a difference of T seconds among them (_i.e._, L\({}_{\text{t}}\pm\) T). Note, that if we include the radio subsystem, the lifetime of every node would be less than L\({}_{\text{t}}\). Moreover, L\({}_{\text{t}}\) will depend on the initial energy stored in the nodes. As can be observed, clustering prolongs the lifetime of the network. Single nodes (clusters having only one node) are the first set of dying nodes (_i.e._, dying at L\({}_{\text{t}}\)). Then, cluster members of clusters of size two nodes die at 1.3 Lt and clusters of size three die at 1.43 L\({}_{\text{t}}\). We may observe, thus, increments of 30% and 43% in the lifetime of the sensors for these cluster sizes. Figure 10:2 Consumed power (P) for a non-cooperative tier/network of nodes consisting of Cyclops. Figure 10:3 Prolongation Lifetime Ratio (PLR) for different node densities in the clustered tier with an overlapping scale equal to (a) 0.5. (b) 0.6, in four states of base awakening period.