# An Energy-Efficient and Robust Multipath Routing Protocol for Cognitive Radio Ad Hoc Networks
## 3 System Model  
We consider a battery-powered CRAHN with \(N\) PUs and \(M\) SUs. The number of licensed channels is assumed to be equal to the number of PUs such that each PU has a particular channel assigned to it. PUs hold undisputable license for specific spectrum bands and do not deviate from their assigned spectrum bands. SUs are expected to exchange their network-related control information through a dedicated common control channel. SUs are assumed to possess two separate radio transceivers for control packets and data packets, respectively. It is also expected that SUs can tune their radio transceivers to any vacant PU channels. The number of vacant PU channels varies from one SU to another and with the passage of time. Therefore, SUs are supposed to have the spectrum sensing capability to determine the vacant channels not in use by PUs and the ability to use them opportunistically. In this paper, we assume that PUs are static whereas SUs can be mobile. The SU mobility that concerns PUs is dealt with by constantly checking if the SU is within the transmission range of PUs. If a SU moves away from the PU in ON state such that it goes beyond transmission range, then the SU is able to use the corresponding channel. On the other hand, if a SU moves into the transmission range of the PU in ON state, then it drops the corresponding channel and switches to the next channel.  
Good PU activity modeling is important to make the environment more realistic. PUs are modeled by virtue of independent and identically distributed ON and OFF process with exponential distribution [24]. Figure 1 shows the alternating ON and OFF states of a PU. In ON state, the PU is active and occupies its channel for period \(T_{ON}\), during which it performs its data transmission via that channel; thus, SUs are not allowed to use the channel. Conversely, in OFF state, the PU is inactive, so SUs can temporarily use the licensed channel until the PU becomes active. The state of PU \(n\) at time \(t\), \(S_{t}(n)\), is active (1) or inactive (0) and can be represented as  
\[S_{t}(n)=\left\{\begin{array}{ll}1&\text{if PU is active}\\ 0&\text{otherwise.}\end{array}\right. \tag{1}\]  
If the ON and OFF periods of a PU are exponentially distributed with mean ON period \(\frac{1}{\alpha}\) and mean OFF period \(\frac{1}{\mu}\), where \(\alpha\) is the departure rate of the PU and \(\mu\) is the arrival rate of the PU, then the probability of the PU being in ON state, \(P_{on}\), is given by  
\[P_{on}=\frac{\mu}{\mu+\alpha} \tag{2}\]  
Obviously, the probability of the PU being in OFF state, \(P_{off}\), is \(1-P_{on}\) and can be represented as  
\[P_{off}=\frac{\alpha}{\mu+\alpha} \tag{3}\]  
On the other hand, \(P_{on}\) and \(P_{off}\) follow an exponential distribution. Given ON period \(u\), therefore, the probability of the PU being in ON state, \(P_{on}(u)\), is given by  
\[P_{on}(u)=1-e^{-\lambda u} \tag{4}\]  
where \(\lambda\) is the mean ON period of the PU (i.e., \(\lambda=\frac{1}{\alpha}\)). Given OFF period \(v\), the probability of the PU being in OFF state, \(P_{off}(v)\), is given by  
\[P_{off}(v)=1-e^{-\gamma v} \tag{5}\]  
where \(\gamma\) is the mean OFF period of the PU (i.e., \(\gamma=\frac{1}{\mu}\)). From Equation (4), the ON period \(u\) of a PU can be represented by  
\[u=\frac{-\ln(1-P_{on}(u))}{\lambda} \tag{6}\]  
Figure 1: Alternating ON/OFF states of primary users (PUs).  
From the Equation (5), the OFF period \(v\) of a PU can be represented by  
\[v=\frac{-\ln(1-P_{off}(v))}{\gamma} \tag{7}\]