# Mobility-Enhanced Reliable Geographical Forwarding in Cognitive Radio Sensor Networks
## 4 MROR System and Network Model
### Mobility-Enhanced Channel Availability Modeling
#### 4.2.1 PU Protection
In line with the analysis presented in [8] and taking into account the fact that a sequence of independent random variables, which are uniformly distributed, can be used to find a joint distribution, the average fraction of the summation of MIPs in the network for a channel will be:  
\[\wp_{i}\left(\delta_{s,i}\right)=1-e^{-\delta_{p}j\pi\left(Z_{e,i}\left( \delta_{s,i}\right)+\ell_{i}\right)^{2}} \tag{8}\]  
Accordingly, the distributions of the duration for the two states (_i.e._, ON and OFF) can be derived thusly: Considering a PU traffic resumption rate of \(\gamma_{busy,i}\) on channel \(i\); the stay time of the ON period follows an exponential distribution [28]:  
\[\tau_{on,i}\sim e^{\left(\gamma_{busy,i}+\frac{v_{m}}{\tau_{c,i}+\tau_{i}} \right)} \tag{11}\]  
From Equation (11), based on the stationary distribution of the ON/OFF states that can be approximated from Equation (5) (_i.e._, \(\rho_{on,i}=1-\wp_{i}\) and \(\rho_{off,i}=\wp_{i}\)), using the detailed balanced equation, \(\rho_{on,i}\gamma_{on,i}=\rho_{off,i}\gamma_{off,i}\), the state transition rate for \(OFF\to ON\), \(\gamma_{off}\), results in:  
\[\gamma_{off,i}=\frac{\wp_{i}\wp_{busy,i}}{1-\wp_{i}\wp_{busy,i}}\left(\gamma _{busy,i}+\frac{v_{n}}{Z_{e,i}+\ell_{i}}\right) \tag{12}\]  
From Equation (5), the stay time for the OFF state is derived:  
\[\tau_{off,i}\sim e^{\left(\gamma_{off,i}\right)} \tag{13}\]  
The sensing interval of an SU on any channel \(i\) depends on both the frequency of the PU activity on that channel and the distance it has traveled after the previous sensing operation [16]:  
\[t_{i}^{*}=\max\left\{T_{s,i},\min\left\{-\frac{\ln\left(1-\zeta\right)}{\gamma _{off}},\frac{\ell_{i}}{v}\right\}\right\} \tag{14}\]  
where \(\zeta\) is a predefined threshold, \(\zeta\) (\(0<\zeta<1\)) that defines the cdf of the channel OFF state. It is assumed that the licensed spectrum is utilized by PUs according to a probabilistic model. Since the PU arrival is independent, the ON and OFF states are exponentially distributed [30]. On arrival, the licensed user stays for \(T_{PU}^{off,i}\) seconds. The ON and OFF states are respectively characterized by the arrival time, \(\tau_{on,i}\) seconds, for the ON state and an absence time, \(\tau_{off,i}\), for the OFF state. The transition of the ON to OFF state is with a probability, \(P_{on,i}\), and it reverses its state (_i.e._, OFF to ON state) with a probability \(P_{off,i}\). \[P_{on,i}=\frac{\tau_{on,i}}{\tau_{on,i}+\tau_{off,i}} \tag{15}\]  
Hence, when \(P_{on}\) is high, more PU activity on the licensed spectrum will be observed, which means fewer opportunities for SU transmission and _vice versa_. The OFF state probability will be:  
\[I_{i}^{U}\left(\delta_{s,i},Z_{e,i}\right)=\frac{2\pi P_{o}d_{o}^{\alpha}\delta _{s,i}}{\alpha-2}\left(Z_{e,i}-Z_{o}\right)^{2-\alpha} \tag{5}\]0  
Once an SU commences data transmission on the data channel, it only releases the channel after it completes sending its packet. Suitable adoption of SU transmission power is used to control any interference that arises thereof [31]. For the sake of simplicity, it is assumed that a specific frequency band having C channels of the same bandwidth is chosen as the sensors' operation area. Furthermore, all control signaling are via a dedicated common control channel (CCC), and the single transceiver is tuned to CCC by default and changes only during the data-forwarding phase. Given that \(T_{s,i}\) represents the time it takes to perform spectrum sensing and \(T_{sw,i}\) represents the switching time to channel \(i\), the average time an SU, \(n\), has access to any channel, \(i\in\mathbb{C}\), will be:  
\[I_{i}^{U}\left(\delta_{s,i},Z_{e,i}\right)=\frac{2\pi P_{o}d_{o}^{\alpha}\delta _{s,i}}{\alpha-2}\left(Z_{e,i}-Z_{o}\right)^{2-\alpha} \tag{5}\]1where \(N_{n,s,i}\) represents the total number of times \(n\) performs spectrum sensing within the channel access time, \(T_{i}\).