## 5 Congestion Mitigation for Bottlenecked Secondary User
Hence, within the time interval \(\tau_{rp}\), \(\theta_{b}^{l}\big{(}\tau_{rp}\big{)}\) of bottlenecked SU \(b\) can be further formulated by:  
\[\theta_{b}^{l}\big{(}\tau_{rp}\big{)}=\sum_{m\in\mathcal{N}}\Xi_{m}(p_{m}(t))- \chi_{b}\big{(}\tau_{rp}\big{)} \tag{39}\]  
where \(\chi_{b}\big{(}\tau_{rp}\big{)}\) is the amount of the data sequence of SSR removed by bottlenecked SU \(b\) within a time interval \(\tau_{rp}\). So far, we have mathematically derived the amount of the data sequence of SSR transmitted from SSNs to bottlenecked SU \(b\) under the condition that the \(m\)-th SSN employs the transmit power \(p_{m}(t)\), for \(m\in\mathcal{N}\). Next, we are concerned with how to mitigate the congestion of bottlenecked SU \(b\) by the aid of the proposed distributed power control framework for IT and MT SSNs. The basic idea of congestion mitigation for bottlenecked SU \(b\) is to alleviate its buffer load because of the accumulated amount of the data sequence of the SSR transmitted from IT and MT SSNs. It is uncovered that the transmit power of the \(m\)-th SSN may have a bearing on \(\Xi_{m}\) from Equation (38). Our objective is to leverage the distributed power control to reduce the amount of bits of the data sequence of the SSR transmitted from the \(m\)-th SSN to bottlenecked SU \(b\), in order to lower the amount of the data sequence of the SSR in the internal buffer of bottlenecked SU \(b\). This operation will further ensure that offered data load does not exceed available buffer capacity of bottlenecked SU \(b\), i.e., \(\phi_{b}\big{(}t_{0}^{\prime}+\tau_{rp}\big{)}<B_{b}\). The key point to achieve congestion mitigation for bottlenecked SU \(b\) is established with a block diagram shown in Figure 4. It is worth remarking that the amount of the data sequence of SSR in the internal buffer of bottlenecked SU \(b\) can be effectively reduced by the proposed distributed power control framework for IT and MT SSNs under the noncooperation and cooperation scenarios. Conceptually, the reduction of the amount of the data sequence of SSR will release the capacity of the internal buffer for bottlenecked SU \(b\), which naturally further attains the congestion mitigation for bottlenecked SU \(b\). In the following, we analyze the impact of optimal transmit power for the \(m\)-th SSN on the reduction of the amount of the data sequence of SSR in the internal buffer of bottleneck SU \(b\) rigorously from the noncooperation and cooperation cases. 1. _Noncooperative Optimal Transmit Power Case:_ In this case, suppose, without loss of generality, that \(\tau_{rp}W\triangleq 1\). By replacing \(p_{m}(t)\) in Equation (38) with \(p_{m}^{NC}(t)\) in Equation (28), \(\Xi_{m}\big{(}p_{m}^{NC}(t)\big{)}\) for the \(m\)-th SSN within a time interval \(\tau_{rp}\) can be expressed as:  
Figure 4: The block diagram of the key point to achieve congestion mitigation for bottlenecked secondary user. \[ \begin{array}{ll}\Xi_{m}\big{(}p_{m}^{NC}(t)\big{)}&=\tau_{rp}W\log_{2}\bigg{(}1+ \frac{p_{m}^{NC}(t)K|h_{m-b}|^{2}}{N_{0}W}\bigg{)}\\ &\approx\log_{2}\bigg{(}\frac{p_{m}^{NC}(t)K|h_{m-b}|^{2}}{N_{0}W}\bigg{)}\\ &\approx\log_{2}\bigg{(}p_{m}^{\max}\bigg{(}1-\frac{H_{m}\big{(}\mathcal{T}_{d }(m)\|T_{f}(m)\big{)}\tau_{rp}}{2(r+1)e^{\delta_{1}Dm\big{(}\mathcal{T}_{d}|^{ 2}P_{f}\big{)}}}\bigg{)}\frac{K|h_{m-b}|^{2}}{N_{0}W}\bigg{)}\end{array}  \tag{40}\]  
By plugging \(\Xi_{m}\big{(}p_{m}^{NC}(t)\big{)}\) into Equation (39), we have:  
\[\theta_{b}^{I}\big{(}\tau_{rp}\big{)}=\underbrace{\sum\limits_{m=1}^{n_{IT}} \Xi_{m}\Big{(}p_{m}^{NC}(t)\Big{)}}_{\text{for $n_{IT}$ STSSN}}+\underbrace{\sum\limits_{m=n_{IT}+1}^{n_{IT}+n_{MT}}\[PERSON]_{m} \Big{(}p_{m}^{NC}(t)\Big{)}}_{\text{for $n_{MT}$ MTSSNN}}-\chi_{b}\big{(}\tau_{rp}\big{)} \tag{41}\]  
Thus, the reduction of the amount of the data sequence of SSR in the internal buffer of bottlenecked SU \(b\), denoted by \(\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}\), is equivalent to:  
\[