## 2 System Model  
In this section, we present the system model, assumptions, and spectrum sensing design. We assume a decentralized interweave TDMA hierarchical CR network, with nodes self-organizing into a network [19; 29; 33]. In the study, we consider a CRN with \(N\) narrowband channels, with each narrowband channel licensed to a PU. It is assumed that a SU employs an ED scheme. The performance metric in term of detection probability and false-alarm probability measures the performance of the ED scheme. On the basis of the probability density function (PDF) of the test statistic, complex-valued phase-shift keying signaling, and circularly symmetric complex Gaussian noise [32; 33; 34], the closed-form expressions of sensing probabilities are considered. The SU is assumed to be equipped with a single narrowband antenna to sense the considered channel. This assumption would be applied to real-world hardware-constrained CRNs. A SU performs sensing procedure in a sequential order, with a fixed sampling frequency and constant transmit power, to locate the spectrum holes. The considered channels are assumed to be idle or busy depending on the sensing outcomes. We assume that the SU transmitter is always ready to transmit the packet, provided that considered channel is available [36]. The secondary network cannot guarantee on-time services, i.e., QoS is not guaranteed. The suitable traffic pattern for the secondary network can be best-effort data. Therefore, we assume that the secondary traffic is back-logged, i.e., the secondary queue always has a data to be transmitted. It is also assumed that the SU has the switching capability to drive multiple spectrum handoffs. We assume that the spectrum sensing procedure is executed until handoff operation is initiated. In this study, spectrum handoff delay is not considered, and is assumed that the handoff is executed immediately when necessary. In addition, the sensing duration is assumed to be identical at each channel. We consider a single set of communicating SU nodes for the system, and that the sensing procedure is performed at each channel. For ease of reference, we summarize our commonly used notations in Table 1.  
The time slots in OSA-based frame structure of SU over different channels is shown in Figure 2. The frame structure consists of sensing time and transmission time over the given set of channels. After the sensing procedure is complete, the SU transmits in the unoccupied channel for the transmission time. The available transmission time in the \(j\)th channel, i.e., after the \((j-1)\)th handoff, \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(P_{FA}\) & Probability of false alarm \\ \(P_{D}\) & Probability of detection \\ \(P_{MD}\) & Probability of miss-detection \\ \(T_{d.1}\) & Transmission duration over first channel \\ \(t_{s}\) & Sensing duration \\ \(P_{ON}\) & PU active state probability \\ \(P_{OFF}\) & PU inactive state probability \\ \(R\) & PU mean inactive duration \\ \(S\) & PU mean active duration \\ \(A\) & RV to define \(OFF\) duration \\ \(B\) & RV to define \(ON\) duration \\ \(P_{S}\) & Packet size \\ \(R_{p}\) & Data rate \\ \(\gamma\) & SNR of PU signal at SU \\ \(\Omega\) & PU deployment density \\ \(N_{s}\) & Number of sensing samples \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of Notations. is \(T_{d.1}-(j-1)\ t_{s}\). It is assumed that if the \(j\)th channel is found free, the SU will be able to transmit the corresponding packet in the available transmission duration, and will not sense the next channel, i.e., \((j+1)\)th channel. In this study, we assume the upper limit for the allowable number of handoffs, i.e., \(N\), and also, that the transmission duration over the considered channel is enough to transmit the corresponding packet. If the \(N\)th channel is found busy, the SU waits for the next time-slot to sense the first channel again.