## 6 Simulation Results  
In this section, we provide simulation results to evaluate the performance of the proposed congestion mitigation approach by using the distributed power control framework for IT and MT SSNs in the rectangular grid based SN-CRN, and investigate the impact of key system parameters on the performance. Particularly, our simulations pay more attention to evaluate the effect of the proposed distributed power control framework for IT and MT SSNs on the reduction of the amount of the data sequence of SSR in the internal buffer of bottleneck SU \(b\). Technically, the reduction of the amount of the data sequence of SSR in the internal buffer will free the buffer capacity of bottleneck SU \(b\), which can bring about the congestion mitigation for bottlenecked SU \(b\). As illustrated in Figure 5, all the simulations are carried out on a rectangular grid topology within a torus area of 100 m \(\times\) 100 m where one bottlenecked SU is randomly placed within the area of IT over three-tier structure. It suffices to mention that the horizontal or vertical distance between any SSNs is initialized to be \(d\) = 20 m. For convenience, the IT SSN and the MT SSN are marked by IT-\(m\) and MT-\(n\) in sequence for \(m\in\mathcal{N}_{IT}\) and \(n\in\mathcal{N}_{MT}\), respectively, as shown in Figure 5. We assume that there are \(N_{c}\) = 12 licensed uplink channels allocated to PUs. The SOP usage probability \(\delta_{b}\) is set to 0.65 for the bottlenecked SU. Also, the total average power constraint assigned to IT and MT SSNs are assumed to \(P_{av}^{IT}\) = 18 dBm and \(P_{av}^{MT}\) = 8 dBm, respectively, to mitigate the interference among IT and MT SSNs. As for the AWGN multiple-access channel, the path-loss exponent \(\kappa\) has been set to 8, and the channel bandwidth used for SSNs is assumed to be 2 MHz according to IEEE 802.15.4a channel model. The constant \(K\) under the channel capacity formulation given in Equation (4) is defined as 0.005. In addition, the noise power spectral density \(N_{0}\) under this channel model is initialized as \(-\)3 dBm. We also adopt the receiving reference power by bottleneck SU \(b\)\(P_{0}\) = 20 dBm under the reference distance \(d_{0}\) = 20 m.  
In all the simulations, the detection and false alarm probabilities for IT and MT SSNs over \(N_{c}=12\) licensed uplink channels have been initialized as in Figure 6. We also set the number of subintervals in Algorithm 1 to be the same value for all SSNs, i.e., \(\xi=\varsigma=8\). We also assume that the minimum value \(X_{d}\) and the maximum value \(Y_{d}\) in derivation of the detection probability distribution are set to 0.6 and 1, respectively. Likewise, the minimum value \(X_{f}\) and the maximum value \(Y_{f}\) in derivation of the false alarm probability distribution are set at 0 and 0.3, respectively. Figure 5: Simulation scenario: the rectangular grid topology with three-tier structure within a torus area of 100 m \(\times\) 100 m. The dotted lines correspond to the transmissions of the spectrum sensing results from interior tier (IT) and middle tier (MT) spectrum sensor nodes to bottlenecked secondary user. SU: secondary user; PU: primary user; ET: exterior tier. Figure 6: The detection and false alarm probabilities for spectrum sensor nodes from interior tier and middle tier over 12 uplink channels. \(\Pi\): interior tier; MT: middle tier; SSN: spectrum sensor node. For simplicity, we deliberately choose some of the IT and MT SSNs in the simulation including IT-1, MT-2, MT-5, and MT-7, to evaluate the performance of our developed approach. Specifically, the detection probability and false alarm probability distributions of the selected SSNs from IT and MT generated from Algorithm 1 are assumed to comply with the corresponding distributions as given by Figure 7.  
The proposed OTPA algorithm for both the NoCoPC problem and the CoPC problem under the distributed power control framework for IT and MT SSNs is compared with the well-known power balancing (PB) algorithm in [35].