## 5 Congestion Mitigation for Bottlenecked Secondary User
\begin{array}{ll}\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}&=\sum\limits_{ m=1}^{n_{IT}}\big{(}\[PERSON]_{m}(p_{m}(t))-\Xi_{m}\big{(}p_{m}^{NC}(t)\big{)}\big{)}+ \sum\limits_{m=n_{IT}+1}^{n_{IT}+n_{MT}}\big{(}\Xi_{m}(p_{m}(t))-\Xi_{m}\big{(} p_{m}^{NC}(t)\big{)}\big{)}\\ &\approx\sum\limits_{m\in\mathcal{N}}\log_{2}\bigg{(}\frac{p_{m}(t)}{p_{m}^{ NC}(t)}\bigg{)}\end{array}  \tag{42}\]  
**Proposition 5**. : _Under the condition of the transmit power \(p_{m}(t)=P_{m}^{\max}\) for the \(m\)-th SSN, \(\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}\) satisfies the following upper bound:_  
\[\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}\leq\sum\limits_{m\in\mathcal{N}} \log_{2}\Bigg{(}1/\Bigg{(}1-\frac{H_{m}\Big{(}\Upsilon_{d}(m)\|\Upsilon_{f}(m )\Big{)}\tau_{rp}}{2(r+1)e^{\delta_{1}\cdot D_{m}(\mathcal{T}_{d}|\mathcal{T}_ {f})}}\Bigg{)}\Bigg{)} \tag{43}\]  
**Proof. ** Since \(p_{m}(t)=P_{m}^{\max}\). Under this condition, using Equation (28), we have:  
\[ \begin{array}{ll}\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}&\approx\sum \limits_{m\in\mathcal{N}}\log_{2}\bigg{(}\frac{p_{m}(t)}{p_{m}^{NC}(t)}\bigg{)} \\ &\leq\sum\limits_{m\in\mathcal{N}}\log_{2}\Big{(}\frac{p_{m}^{\max}}{p_{m}^{ NC}(t)}\Big{)}\\ &=\sum\limits_{m\in\mathcal{N}}\log_{2}\bigg{(}1/\Bigg{(}1-\frac{H_{m}\big{(} \Upsilon_{d}(m)\|\Upsilon_{f}(m)\big{)}\tau_{rp}}{2(r+1)e^{\delta_{1}Dm(\mathcal{ T}_{d}|\mathcal{T}_{f})}}\bigg{)}\bigg{)}\end{array}  \tag{44}\]  
Then we obtain the upper bound in Equation (43), thus completing the proof. \(\square\)
2. _Cooperative Optimal Transmit Power Case_: In this case, with the assumption \(\tau_{rp}W\triangleq 1\), by replacing \(p_{m}(t)\) in Equation (38) with \(p_{m}^{\mathbb{C}}(t)\) in Equation (34), \(\Xi_{m}\big{(}p_{m}^{\mathbb{C}}(t)\big{)}\) for the \(m\)-th SSN within a time interval \(\tau_{rp}\) can be given by: \[\begin{array}{ll}\Xi_{m}\big{(}p_{m}^{\mathbb{C}}(t)\big{)}&=\tau_{rp}W\log_ {2}\bigg{(}1+\frac{p_{m}^{\mathbb{C}}(t)K|h_{m-b}|^  
By substituting \(\Xi_{m}\big{(}p_{m}^{C}(t)\big{)}\) into Equation (39), we obtain:  
\[\theta_{b}^{I}\big{(}\tau_{rp}\big{)}=\underbrace{\sum\limits_{m=1}^{n_{IT}}\Xi_ {m}\Big{(}p_{m}^{C}(t)\Big{)}}_{\text{for }n_{IT}\text{IT}\text{SSNs}}+\underbrace{\sum\limits_{m=n_{IT}+1}^{n_{IT}+n_ {MT}}\Xi_{m}\Big{(}p_{m}^{C}(t)\Big{)}}_{\text{for }n_{MT}\text{MT}\text{SSNs}}-\chi_{b}\big{(}\tau_{rp} \big{)} \tag{46}\]  
Therefore, similar to Equation (42), \(\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}\) is also approximately formulated as:  
\[\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}\approx\sum\limits_{m\in\mathcal{N }}\log_{2}\bigg{(}\frac{p_{m}(t)}{p_{m}^{C}(t)}\bigg{)} \tag{47}\]  
**Proposition 6**. : _Under the condition of the transmit power \(p_{m}(t)=P_{m}^{\max}\) for the \(m\)-th \(SSN\),\(\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}\) satisfies the following upper bound:_  
\[\Delta\theta_{b}^{I}\big{(}\tau_{rp}\big{)}\leq\sum\limits_{m\in\mathcal{N}} \log_{2}\Bigg{(}1/\Bigg{(}1-\frac{H_{m}\Big{(}\Upsilon_{d}(m)\|\Upsilon_{f}(m )\Big{)}\tau_{rp}}{2(r+1)\sum\limits_{m\in\mathcal{N}}e^{\delta_{b}\cdot D_{m} (\mathcal{P}_{d}\|\mathcal{P}_{f})}}\Bigg{)}\Bigg{)} \tag{48}\]  
Proof. : Similar to the proof of Proposition 5, it is easy to identify the upper bound in Equation (48), thus completing the proof. \(\square\)