## 5 Cooperative Scheduling for Object Detection in Cluster-Based WMSN.
### Cluster-based \(T_{p}\) and \(T_{interval}\) Computation  
Let us consider as baseline mechanism a non-collaborative duty-cycled scheme in which every node awakes with an interval period of time T and senses the area (_i.e._, takes a picture and performs object \begin{table}
\begin{tabular}{l} \hline \hline \multicolumn{1}{c}{**The Algorithm**} \\ \hline
1: **For all** C\({}_{\text{j}}\) // all clusters in parallel // \\
2: i = 0 // start with the first member of each cluster // \\
3: Wake up member number i \\
4: Capture an image and then call object detection \\
5: **If** (detection==true) \\
6: Send the image to sink \\
7: **End-If** \\
8: Delay (Tinterval) // each cluster has a proprietary Tinterval // \\
9: i = i + 1(mod Csize) // select next node of the cluster // \\
10: **Goto 3** \\
11: **End-For** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cluster-based Cooperative Scheduling for Object Detection. Figure 9: (a) Period of awakening a given node in the un-cooperative scheduling. (b) Scheduling for a cluster consisting of three members (S\({}_{1}\), S\({}_{2}\), S\({}_{3}\)). detection) as tier 1 in [15]. The objective of the proposed collaborative mechanism is to produce a cluster-based duty-cycled scheduler in which: (i) Each node is awakened and senses the area with a reliable period of T\({}_{\rm p}\)\(>\) T taking advantage of the overlapping among nodes in the cluster, thus, saving energy and increasing network lifetime. Each cluster will have its own T\({}_{\rm P}\) interval, determined according to the cluster-size and the overlapping scale. (ii) During the sleeping period of each member of a given cluster, other nodes belonging to the cluster are awakened with intervals of T\({}_{\rm interval}\)\(<\) T (that is equal to: T\({}_{\rm p}\)/C\({}_{\rm size}\)) in a sequential manner. The area sensed by each cluster is related to the MCCD area. In order to compute T\({}_{\rm p}\) we will consider the MCCD area. By awaking each member of a given cluster, in average, a part of the related MCCD with a ratio equal to 1/\(\beta\) is captured (Equation (11)). Note that the MCCD is an area of \(\beta\).A\({}_{\rm Fov}\) and is sensed by C\({}_{\rm size}\) overlapping members, thus sensing the environment by each node delivers information not only from the FoV of the awakened node but also from some overlapped parts of the FoV of other nodes in the same cluster. Then, we may define the node interval duty-cycle period as:  
\[\rm T_{P}=T\cdot\frac{C_{size}}{\beta}=T\cdot\frac{C_{size}}{C_{size}-\gamma \cdot(C_{size}-1)} \tag{12}\]  
Note that the T\({}_{\rm P}\) is proprietary for each cluster in terms of its cluster-size and overlapping scale. As it was mentioned before, the MCCD calculated by Equation (10) is the maximum covering domain of a cluster while the effective cluster covering domain may be less than MCCD since some members of a given cluster may overlap more than the region determined by \(\gamma\). Consequently, a given cluster can cover an area less than \(\beta\)\(\cdot\)A\({}_{\rm FoV}\). Thus, using \(\beta\) gives us the lowest interval T\({}_{\rm p}\) and thus the most reliable one since lower values of \(\beta\) would increase the interval T\({}_{\rm P}\). On the other hand, members of a cluster are awakened sequentially to sense their environment in an intermittent way with time intervals equal to T\({}_{\rm interval}\):  
\[\rm T_{interval}=\frac{T_{P}}{C_{size}}=\frac{T}{C_{size}-\gamma\cdot(C_{size} -1)}\leq T \tag{13}\]  
Let us consider Figure 9(b) and for example a cluster with three members, C = {S\({}_{1}\), S\({}_{2}\), S\({}_{3}\)}, cluster-head S\({}_{1}\) and \(\gamma\) = 0.5. Every node will be awakened every T\({}_{\rm P}\) = 1.5 T seconds and the area will be monitored every T\({}_{\rm interval}\) = 0.5 T seconds. As can be observed, every sensor is awakened with a period higher than the non-collaborative scheme but the area is monitored more times. Then, the area duty-cycled frequency is increased while the sensor duty-cycled frequency is reduced.