# Performance of Dynamic Time Division Broadcast Protocol with Rateless Coding
## 3 Performance Analysis
### Outage Probability and Expected Rate
Then, for the target data rate \(C\) bits/s/Hz, according to the total probability formula, the outage probability of DTDBC-RC is  
\[P_{out}=\underbrace{P_{r}\left(\Theta_{S_{1}S_{2}}^{1}\cap\left(\Theta_{S_{1} R}^{1}\cup\Theta_{S_{2}R}^{2}\right)\right)}_{\Sigma_{1}}+\underbrace{P_{r}\left( \Theta_{S_{1}S_{2}}^{1}\cap\overline{\Theta_{S_{1}R}^{1}}\cap\overline{\Theta _{S_{2}R}^{1}}\left(\Theta_{S_{1}RS_{2}}\cup\Theta_{S_{2}RS_{1}}\right)\right) }_{\Sigma_{2}}, \tag{5}\]  
where \(\Theta_{iRj}\) (with \(i,j\in\left\{S_{1},S_{2}\right\}\) and \(i\
eq j\)) is the event that the source \(j\) still cannot decode successfully after combining the messages received from both the source \(i\) and the relay \(R\), i.e.,  
\[\Theta_{iRj}=\left\{\left(h_{ij},h_{jR}\right):\alpha\log\left(1+\rho\left|h_{ ij}\right|^{2}\right)+\left(1-2\alpha\right)\log\left(1+\rho\left|h_{jR} \right|^{2}\right)<\left(1-\alpha\right)C\right\}\]  
and  
\[\Theta_{S_{1}S_{2}}^{1}=\left\{h_{S_{1}S_{2}}:\alpha\log\left(1+\rho\left|h_{ S_{1}S_{2}}\right|^{2}\right)<\left(1-\alpha\right)C\right\},\]  
\[\Theta_{S_{1}R}^{1}=\left\{h_{S_{1}R}:\alpha\log\left(1+\rho\left|h_{S_{1}R} \right|^{2}\right)<\left(1-\alpha\right)C\right\},\]  
\[\Theta_{S_{2}R}^{2}=\left\{h_{S_{2}R}:\alpha\log\left(1+\rho\left|h_{S_{2}R} \right|^{2}\right)<\left(1-\alpha\right)C\right\}.\]  
With the identity (for any set A, B, and C)  
\[P_{r}\left(A\cap\left(B\cup C\right)\right)=P_{r}\left(A\cap B\right)+P_{r} \left(A\cap C\right)-P_{r}\left(A\cap B\cap C\right), \tag{6}\]  
the first and second terms at the right-hand side (RHS) of Equation (5) can then be calculated as  
With the outage probability given in Theorem 1, the expected rate--defined as the expected successfully transmitted throughput--of DTDBC-RC is expressed as (according to the total probability formula)  
\[ \begin{split} C_{DTDBC-RC}&=\underbrace{\int_{z_{0}}^{ +\infty}\log(1+\rho x)\frac{1}{\Omega_{0}}e^{-x/\Omega_{0}}dx}_{\Sigma_{3}}+ \underbrace{2\left(1-\alpha\right)C\left(1-e^{-z_{0}/\Omega_{0}}\right)\left(1- P_{out}\right)}_{\Sigma_{4}}\\ &=\frac{1-\alpha}{\alpha}Ce^{-z_{0}/\Omega_{0}}+\frac{\rho}{ \ln 2}\int_{z_{0}}^{+\infty}\frac{e^{-x/\Omega_{0}}}{1+\rho x}dx+2\left(1-\alpha \right)C\left(1-e^{-z_{0}/\Omega_{0}}\right)\left(1-P_{out}\right),\end{split}  \tag{9}\]  
where \(\Sigma_{3}\) and \(\Sigma_{4}\) at the RHS denote the expected rates corresponding to State 2 and State 1, respectively, and \(z_{0}=\left(2^{\left(1-\alpha\right)C/\alpha}-1\right)/\rho\) is the channel gain power when \(S_{1}\) (or \(S_{2}\)) decodes successfully exactly at \(\alpha T\). Similarly, the expected rates of TDBC-IR and sub-DTDBC-RC are derived as follows for comparison with DTDBC-RC. \[ \begin{split} C_{TDBC-IR}&=\frac{2M}{\alpha WT} \left(1-P_{direct}\right)+\frac{2M}{WT}P_{direct}\left(1-P_{out}\right)\\ &=\frac{1-\alpha}{\alpha}Ce^{-z_{0}/\Omega_{0}}+2\left(1-\alpha \right)C\left(1-e^{-z_{0}/\Omega_{0}}\right)\left(1-P_{out}\right),\end{split}  \tag{10}\]  
\[ \begin{split} C_{sub-DTDBC-RC}&=\sum\limits_{i=0}^{N- 1}\frac{2M}{\alpha TW\left(1+i\right)/N}\left(P_{r}^{i}-P_{r}^{i+1}\right)+2 \left(1-\alpha\right)C\left(1-e^{-z_{0}/\Omega_{0}}\right)\left(1-P_{out} \right)\\ &=\sum\limits_{i=0}^{N-1}\frac{\left(1-\alpha\right)CN}{\alpha \left(1+i\right)}\left(P_{r}^{i}-P_{r}^{i+1}\right)+2\left(1-\alpha\right)C \left(1-e^{-z_{0}/\Omega_{0}}\right)\left(1-P_{out}\right),\end{split}