## 4 Adaptive Transmission Strategy Design and Outage Probability Analysis
### Adaptive Transmission Strategy Design  
Based on the definition of the function \(f(P_{T})\), we can know that the network can switch between the BRT and BDT modes. Under the equal power allocation, \(P_{1}=P_{2}=\frac{1}{2}P_{T}\) can be obtained. In such a case, \(f(P_{T})\) can be expressed as  
\[\begin{split} f(P_{T})=&\frac{1}{2}\text{log}_{2} \Bigg{(}1+\frac{(1-\rho)\rho\eta\frac{1}{2}P_{T}|f_{b}|^{2}|g_{b}|^{2}}{(1- \rho)\rho\eta P_{P}|f_{b}|^{2}|l_{r}|^{2}+(1-\rho)P_{P}|l_{1}|^{2}+N_{1}} \Bigg{)}\\ &+\frac{1}{2}\text{log}_{2}\Bigg{(}1+\frac{(1-\rho)\rho\eta\frac {1}{2}P_{T}|f_{b}|^{2}|g_{b}|^{2}}{(1-\rho)\rho\eta P_{P}|g_{b}|^{2}|l_{r}|^{ 2}+(1-\rho)P_{P}|l_{2}|^{2}+N_{2}}\Bigg{)}\\ &-\text{log}_{2}\Bigg{(}1+\frac{\frac{1}{2}P_{T}h^{2}}{P_{P}l_{ i}^{2}+\sigma_{i}^{2}}\Bigg{)}-\text{log}_{2}\Bigg{(}1+\frac{\frac{1}{2}P_{T}h^{2}}{P _{P}l_{i}^{2}+\sigma_{i}^{2}}\Bigg{)}\end{split} \tag{12}\]  
In order to achieve the adaptive transmission conveniently, with (12), we design a TTPT determining algorithm to find the TTPT \(P_{th}\), which lets the \(R_{T}^{s}\) be equal to the \(R_{T}^{d}\), namely, \(f(P_{th})=R_{T}^{s}(P_{th})-R_{T}^{d}(P_{th})=0\). At the same time, with the dichotomy and the zero-point theorem, we can find the solution of \(f(P_{th})=0\). Based on it, we can just compare the network total transmit power \(P_{T}\) with \(P_{th}\) to determine the network final transmission mode. Specifically, for \(P_{T}\in(P_{T_{1}},P_{T_{u}})\), if \(f(P_{T_{1}})f(P_{T_{u}})<0\), then the threshold \(P_{th}\in(P_{T_{1}},P_{T_{u}})\) that makes \(f(P_{th})=0\) can be found. In such a case, we set \(P_{T_{m}}=\frac{P_{T_{m}}+P_{T_{l}}}{2}\) and calculate \(f(P_{T_{m}})\). Then, if \(f(P_{T_{1}})f(P_{T_{m}})<0\), a threshold \(P_{th}\in(P_{T_{1}},P_{T_{m}})\) that makes \(f(P_{th})=0\) can be found. Otherwise, if \(f(P_{T_{m}})f(P_{T_{u}})<0\), a threshold \(P_{th}\in(P_{T_{m}},P_{T_{u}})\) that makes \(f(P_{th})=0\) can be found. Repeat the above algorithm until \(P_{th}\in(P_{T_{1}},P_{T_{u}})\), where \((P_{T_{1}},P_{T_{u}})<\tau\). From this method, we can find a TTPT \(P_{th}\) that makes \(R_{T}^{s}(P_{th})=R_{T}^{d}(P_{th})\). Based on the above analysis, the detailed algorithm can be seen as Algorithm 1.  
According to Algorithm 1, \(P_{th}\) can be found. With \(P_{th}\), the adaptive bidirectional transmission strategy can be expressed as follows.  
**Case 1.** When \(P_{T}>P_{th}\), \(R_{T}^{s}>R_{T}^{d}\) can be obtained, which means the total transmission rate of BRT is bigger than BDT, and the network still use BRT for signal transmission.  
**Case 2.** When \(P_{T}\leq P_{th}\), \(R_{T}^{s}\leq R_{T}^{d}\) can be obtained, which means the total transmission rate of BDT is not smaller than BRT. Then, the relay node will change to dormant, and the network changes to using BDT for signal transmission.  
```
Input:\(P_{T_{n}}=0\), \(P_{T_{m}}=0\), \(P_{T_{l}}=0\), \(n=0\), \(\tau=10^{-5}\), where \(n\) is the iteration number and \(\tau\) is the convergence factor.
1 while\(f(P_{T_{n}})/f(P_{T_{n}})<0\)do
2\(n=n+1\), \(P_{T}=P_{Tm}\);
3 if\(f(P_{T_{l}})f(P_{T_{m}})<0\);
4 then
5\(P_{th}\in(P_{T_{l}},P_{T_{m}})\);
6
7 end if
8 if\(f(P_{T_{n}})f(P_{T_{m}})<0\);
9 then
10\(P_{th}\in(P_{T_{m}},P_{T_{n}})\);
11
12 end if
13 Repeat the dichotomy;
14 When \((P_{Tl},P_{Tu})<\tau\), \(P_{th}\in(P_{Tl},P_{Tu})\) can be obtained
15
16 end while
```  
**Algorithm 1**TTPT determining algorithm.