## 3 Sum Utilization of Spectrum  
In this section, we consider different PU and SU co-existing network topologies and describe the formulation and analysis of sum utilization of spectrum for different sets of channels. We consider two topological models, i.e., topology-1, and topology-2, as shown in Figure 4, to investigate both the spatial and temporal variation conditions on the set of idle channels between the communicating SUs [30]. The sum utilization of spectrum is evaluated for both topological models separately. Topology-1  
Figure 3: Activity model for the PU. considers the identical sets of idle channels for the communicating SUs, because both SUs are within the coverage range of the same PU. As shown in Figure 4, the communicating SUs, i.e., _SU-A\({}_{Tx}\)_ and _SU-A\({}_{Rx}\)_, or the _SU-B\({}_{Tx}\)_ and _SU-B\({}_{Rx}\)_, lie within the communication range of the same primary network (PRN). Hence, the variation conditions on the set of idle channels is temporal only. For topology-2, communicating SUs are subjected to activity from a possibly different set of PUs. The PU coverage range is comparable to that of the SUs. As shown in Figure 4, the communicating SUs, i.e., _SU-A\({}_{Tx}\)_ and _SU-A\({}_{Rx}\)_, are within the communication range of different PRNs, i.e., PRN-3 and PRN-2, respectively. Hence, the variations of the idle channels are both spatial and temporal. The considered channel \(j\) can be found busy under two scenarios: the PU state is _ON_ and the state is correctly detected, and the PU state is _OFF_ and a false alarm occurs. The probability that the channel \(j\) is found busy is expressed as  
\[P_{B}^{j}=P_{ON}P_{D}^{j}+P_{OFF}P_{FA}^{j} \tag{2}\]  
Furthermore, the channel \(j\) can be found idle under two scenarios: the PU state is _OFF_ and the state is correctly detected, and the PU state is _ON_ and is not detected. The probability that channel \(j\) is found idle is expressed as  
\[P_{F}^{j}=P_{OFF}(1-P_{FA}^{j})+P_{ON}P_{MD}^{j} \tag{3}\]  
Under the two-state model, \(H_{0}\) and \(H_{1}\) denote the events to consider the _OFF_ and _ON_ states, respectively, of the PU in the current channel. Given Equation (3), channel \(j\) in a given time-slot can also be regarded as idle because of a missed detection. The actual free channel \(j\), without a false alarm, is estimated. Thus, the probability of detecting the spectrum opportunity, for the current channel \(j\), is expressed as  
\[P_{D,SO}^{j}=P_{OFF}(1-P_{FA}^{j}) \tag{4}\]  
Figure 4: PU and SU co-existing network topologies: (**a**) topology-1 model; (**b**) topology-2 model. After detecting the spectral opportunity correctly, the communicating SUs require a successful transmission condition. The successful transmission of a packet depends on the activity of PU, i.e., the PU must not access the licensed spectrum band during the transmission of SU packet. Hence, the probability of no activity during the sensing and transmission periods for a single-channel scenario is defined as  
\[P_{T}^{1}=Pr\Big{(}A\geq t_{\text{s}}+(P_{s}^{1}/R_{p})\Big{)} \tag{5}\]  
\[P_{T}^{1}=\int\limits_{t_{\text{s}}+P_{s}^{1}/R_{p}}^{\infty}\frac{1}{R}e^{-t/ R}dt=e^{[(-\frac{1}{k})(t_{\text{s}}+\frac{P_{s}^{1}}{R_{p}})]} \tag{6}\]  
Similarly, the probability of no activity during the sensing and transmission periods, for the \(j\)th channel among a set of \(N\) channels, is defined as  
\[P_{T}^{j}=\int\limits_{t_{\text{s}}+P_{s}^{j}/R_{p}}^{\infty}\frac{1}{R}e^{-t /R}dt=e^{[(-\frac{1}{k})(t_{\text{s}}+\frac{P_{s}^{j}}{R_{p}})]} \tag{7}\]  
where \(P_{s}^{1}\), and \(P_{s}^{j}\), denote the packet size (in bits) in the first, and the \(j\)th channels, respectively. In this study, we assume that the number of bits \(P_{s}^{1}\) are transmitted in the available transmission time \(T_{d.1}\) of the first channel. After the first handover, \(P_{s}^{2}\) (i.e., \(P_{s}^{1}\frac{T_{d.1}-t_{\text{s}}}{T_{d.1}}\)) bits are transmitted in the available transmission time \(T_{d.1}-t_{\text{s}}\) of the second channel.