## 3 Sum Utilization of Spectrum
Moreover, \(P_{s}^{j}\) (i.e., \(P_{s}^{1}\frac{T_{d.1}-(j-1)t_{\text{s}}}{T_{d.1}}\)) bits are transmitted on the \(j\)th channel, after the \(j-1\) handover, in the available transmission time \(T_{d.1}-(j-1)t_{\text{s}}\). If the transmission duration is reduced after handoff, the number of transmitted bits is also reduced with the ratio of transmission duration. A channel error [40] occurs when the SNR of a received packet is low because of path loss or a deep fading. The probability of obtaining an error-free packet over the \(j\)th channel, because of channel errors only for the cognitive transmission, can be expressed as  
\[P_{e-frac}^{j}=(1-BER(\gamma_{s}))^{P_{s}^{j}} \tag{8}\]  
where \(\gamma_{s}\) is the SNR of the secondary transmission (i.e., transmitted power), and \(BER\) is the bit error rate for BPSK in the Rayleigh fading channel and can be expressed as [40]  
\[BER(\gamma_{s})=\frac{1}{4\gamma_{s}} \tag{9}\]  
We assume that the probability of an error-free transmission of packets that are transmitted in the case of a missed detection is one because the primary and secondary packets collide. In this study, we use the probability \(P_{UT.PU}=P_{ON}P_{D}\) for the PU utilization over the given channel set. The probability of sum utilization of spectrum for topology-1 depends on the probability of detecting the spectrum opportunity at the transmitting and receiving SU nodes, probability of no activity during transmission, probability of error-free transmission, and the PU's own utilization probability. From Equations (4), (6), and (8), and PU utilization probability, the probability of sum utilization of spectrum for a single-channel scenario under topology-1 can be expressed as  
\[P_{B}^{j}=P_{ON}P_{D}^{j}+P_{OFF}P_{FA}^{j} \tag{2}\]0  
For the multi-channel scenario, the number of bits in a packet is reduced in each try after handoff; hence, the weighting factor is introduced to consider the utilization as per the transmitted number of bits in each channel. The weighting factor \(W^{1}=\frac{T_{d.1}-t_{\text{s}}}{T_{d.1}}\) is considered for the second channel because \(P_{s}^{1}\frac{T_{d.1}-t_{\text{s}}}{T_{d.1}}\) bits are transmitted in the available transmission time. Moreover, the weighting factor \(W^{j}\ =\ \frac{T_{d,1}-(j-1)t_{s}}{T_{d,1}}\) is considered for the \(j\)th channel, after the \(j-1\) handoff, because \(P_{s}^{1}\frac{T_{d,1}-(j-1)t_{s}}{T_{d,1}}\) bits are transmitted in the available transmission time. Hence, the probability of sum utilization of spectrum, with the set of \(N\) channels under topology-1, can be expressed as  
\[P_{B}^{j}=P_{ON}P_{D}^{j}+P_{OFF}P_{FA}^{j} \tag{2}\]1  
Next, we investigate the sum utilization of spectrum under topology-2. Let \(T\) be the union region of all those PRNs having coverage areas of the communicating SUs. In such a case, the probability of sum utilization of spectrum for topology-2 must be characterized for the two scenarios \(S1\) and \(S2\). \(S1\) refers to a scenario in which no PU is present within region \(T\). \(S2\) refers to a scenario in which there are \(k\) PUs present within region \(T\). We assume that PUs are deployed with a uniform distribution with density \(\Omega\). Thus, the probability that no PU is present within region \(T\) can be expressed as [34]  
\[P_{B}^{j}=P_{ON}P_{D}^{j}+P_{OFF}P_{FA}^{j} \tag{2}\]2  
For the \(S1\) case, the probability of detecting the spectrum opportunity at both SU communicating nodes, and the probability that no PU is present within region \(T\), are considered for the sum utilization of spectrum for the SU. From Equations (4), (8) and (12), the probability of sum utilization of spectrum for the SU, for a single-channel scenario under \(S1\) of topology-2, is expressed by  
\[P_{B}^{j}=P_{ON}P_{D}^{j}+P_{OFF}P_{FA}^{j} \tag{2}\]3  
For the \(S2\) scenario, assume that \(k\)_PUs_, and _PUs.OFF_, are the required events that represent the \(k\) PUs present within region \(T\), and those PUs have an _OFF_ state during SU transmission.