## 4 Cluster-Based Distributed Compressed Sensing for QoS Routing
### Event-Driven Clustering
#### 4.1.1 Problem Formulation
Thus, how to select a cluster head to transmit compressed data without affecting PUs' activities is of significance in CVSNs. To characterize dynamics of channel availability, we model the occupation time of PUs in each data channel as an independent and exponentially distributed alternating ON/OFF random process [11; 33] with birth rate \(b\) and death rate \(a\). The ON state indicates that the channel is occupied by PUs, whereas the OFF state implies that the channel is idle. With the PU activity model, the probability of channel \(c\) being occupied and idle are defined as \(u_{on}^{c}=\frac{b}{a+b}\) and \(u_{off}^{c}=\frac{a}{a+b}\). A larger \(u_{off}^{c}\) indicates that \(c\) is better and more suitable for data transmission. It is assumed that each node independently estimates PUs' activities using the above ON/OFF model instead of relying on a geo-location database [34]. Eligible nodes within the scope of an event participate in DCS. For cluster head election, it is necessary to separate transmission and compression as much as possible. The goal is to balance energy consumption and reduce the probability of congestion. This corresponds to the following optimization problem:  
\[v_{i}^{*}=\text{arg}\ \underset{v_{i}\in\text{Y}}{\text{max}}E_{i}, \tag{4}\]  
subject to:  
\[d(v_{i},s)<d(e,s), \tag{5}\]\[d(v_{i},e)<r, \tag{6}\] \[u_{off}^{c_{i}}\geq U,\] (7) \[\beta-\pi<\arctan\left(\frac{y_{i}-y_{0}}{x_{i}-x_{0}}\right)<\beta. \tag{8}\]  
As shown in objective (4), the node with maximum remaining energy will be elected as a cluster head while satisfying constraints (5)-(8). Constraint (5) ensures that the distance between the cluster head and the sink is less than the distance between the sink and the event. Constraint (6) demands that the distance between the location of a cluster head and the event is less than the sensing radius of a video sensor. Constraint (7) guarantees that the probability of the channel being idle must be larger than a given channel availability threshold. Constraint (8) ensures that a chosen cluster head and the sink should be located on the same side of the event movement locus, where \((x_{0},y_{0})\) is the location of event \(e\) whose moving direction is \(\beta\) and \((x_{i},y_{i})\) is the coordinate of \(v_{i}\). Equation (8) is composed of the inverse tangent function of \(\frac{(y_{i}-y_{0})}{(x_{i}-x_{0})}\), which restricts the area of cluster head selection. We provide a probabilistic guarantee of cluster head selection, in which the probability that the channel being idle should not be below \(\gamma\), expressed by  
Figure 4:0  
where \(U\) is channel availability threshold. Let \(f_{off}^{c_{i}}\) be \(1-u_{off}^{c_{i}}\). Equation (9) can be changed to  
Figure 4:1  
By applying the Markov's inequality on Equation (10), we have  
Figure 4:2  
and  
Figure 4:3  
Comparing (10) and (11), if the following inequation holds,  
Figure 4:4  
probabilistic guarantee inequation (10) for cluster head selection can be satisfied. Inequation (13) can also be expressed as  
Figure 4:5  
and, on this basis, channel availability threshold can be set to  
Figure 4:6  
from which constraint (7) is obtained. The probabilistic guarantee we provide can ensure the probability of the channel being idle as being more significant than a given channel availability threshold. With this method, the chosen cluster head can avoid collision with PUs' activities as much as possible.