## 5 Utilization of Spectrum (UoS) Scheme  
Based on the proposed spatial-temporal spectral hole-sensing model in Section 3, we present an analytical formulation and evaluation of the UoS scheme for the FD-SU TXs deployed at different spatial positions in CR-IoT spectrum-heterogeneous environment. The UoS performance was evaluated by determining the average number of sensing slots, (\(\tau\)), used for the successful secondary communication in each time-slotted frame. Because a dense small-cell scenario is considered, i.e., \(r_{0}<<D1\), the UoS performance was evaluated for two possible cases, as shown in Figure 5. The casesare: (i) the FD-SU TX is present in \(R1\) (FD-SU pair is completely inside the transmission range of PU) and (ii) the FD-SU TX is present in \(R2\) (FD-SU pair is completely outside the transmission range of PU). As explained earlier, the first sensing slot in each time-slotted frame is always HD, i.e., FD-SU TX only performs the sensing procedure. If the PU is not detected at the end of HD sensing slot, the FD-SU TX initiates its transmission and sensing simultaneously. If the PU is detected at the end of HD (or any FD) sensing slot, the FD-SU TX do not transmit (or stops transmission) until the next cycle (time-slotted frame).  
The probability of detecting the spatial-temporal spectral holes during HD and FD sensing slots depends on the sensing outcome, the PU activity, and the spatial position of FD-SU TX. The probability of detecting the spatial-temporal spectral holes in only first \(k\) number of sensing slots in each time-slotted frame can be defined as  
\[P_{ST-h}\left(k\right)=P_{A}\left(k\right)-P_{B}\left(k+1\right). \tag{28}\]  
Here, \(P_{A}\left(k\right)\) refers to the possible scenarios where spatial-temporal spectral holes are detected in greater or equal to \(k\) number of sensing slots. Similarly, \(P_{B}\left(k+1\right)\) refers to the possible scenarios where spatial-temporal spectral holes are detected in greater or equal to \(k+1\) number of sensing slots. Hence, \(P_{A}\left(k\right)\) can be expressed as  
\[P_{A}\left(k\right)=\left(1-P_{MiST}^{0}\right)\sum_{j=1}^{k}\left(1-P_{MiST}^ {1}\right)^{j}. \tag{29}\]  
The probabilities of missing the availability of spatial-temporal spectral holes (\(P_{MiST}^{0}\), and \(P_{MiST}^{1}\)) for FD-SU TX in \(R2\) are determined by each state of the PU, and involve both temporal and spatial (excessive) false alarms. Conversely, \(P_{MiST}^{0}\) and \(P_{MiST}^{1}\) for FD-SU TX in \(R1\) involve only temporal false alarms. It is assumed that all FD-SU TXs are synchronous to their corresponding receivers, and initiates transmission and reception at the same time.  
A channel error occurs when the received SNR, because of path loss or deep fading, falls below the considered threshold [5; 13]. We consider the same bit error rate (BER) in each time-slotted frame. Hence, the number of transmitted bits is adjusted as per the \(k\) number of sensing slots in each time-slotted frame. The probability of obtaining an errored secondary packet, owing to channel errors, can be expressed as [5]  
\[P_{E}=1-\left[\left(1-BER\left(\gamma_{s}\right)\right)^{kPS}\right], \tag{30}\]  
Figure 5: Considered cases of FD-SU TXs.  
where \(kPS\) is the packet size, i.e., transmitted number of bits in \(k\) number of sensing-slots, \(\gamma_{s}\) is the SNR of the secondary transmission, and BER is the bit error rate for the considered secondary channel, and can be expressed as [5]  
\[BER\left(\gamma_{s}\right)=\frac{1}{4\gamma_{s}}. \tag{31}\]  
In our work, we assume that the UoS is contributed to only during the sensing slots during which FD-SU TXs detect the state of the considered channel correctly. Moreover, the access contention for the FD-SU TXs is also not considered.  
From Equations (28)-(30), the average number of sensing slots used for a successful secondary communication in each time-slotted frame can be defined as  
\[\tau=\sum_{k=1}^{\infty}kP_{ST-h}\left(k\right)\left(1-P_{E}\left(k\right)\right) \tag{32}\]