## 5 Performance Analysis
### Average Packet Delay  
In order to derive the average packet delay of RBCB scheme, we model the packet transmission process on each node as a Markov chain. Let \(L\) represent the length of the first-input-first-output (FIFO) buffer on each node. \(U_{i}\) is used to describe the state of buffer, i.e., \(i\) packets are waiting in the buffer. The total number of broadcasting is \(V\), which means a packet exists in the buffer \(V\) frames before  
Figure 7: Two-hop regions of the source and the helper.  
it is cleared. We set the observation point at the end of the considered node's slot every \(V\) frames. Therefore, the buffer has \(L\) states (\(U_{i}0\leq i\leq L-1\)). The one step transition matrix \(\{B_{T}\}_{ij}\) of the buffer state is given by (21), where \(p_{r}\) is the probability of \(r\) packet arrivals between two adjacent observation points. This Markov chain has the unique stationary distribution [41]. \(\omega_{i}\) is used to denote the steady state probability of \(U_{i}\).  
\[B_{T}=\left(\begin{array}{cccccc}p_{0}+p_{1}&p_{2}&p_{3}&\cdots&p_{L-1}&1- \sum\limits_{\begin{subarray}{c}r=0\\ L-2\\ \end{subarray}}^{L-1}p_{r}\\ p_{0}&p_{1}&p_{2}&\cdots&p_{L-2}&1-\sum\limits_{\begin{subarray}{c}r=0\\ L-2\\ \end{subarray}}^{L-2}p_{r}\\ 0&p_{0}&p_{1}&\cdots&p_{L-3}&1-\sum\limits_{\begin{subarray}{c}r=0\\ L-3\\ \end{subarray}}^{L-1}p_{r}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&p_{0}&1-p_{0}\end{array}\right) \tag{21}\]  
We model the packet generation process as a Poisson process. The probability of generating \(r\) packets during a period \(t\) is given by:  
\[p(r,t)=\frac{(\lambda t)^{r}e^{-\lambda t}}{r!},r=0,1,2,\cdots. \tag{22}\]  
where \(\lambda\) is the number of packet arrivals per unit time. Let \(R\) denote the number of packet arrivals during \(V\) consecutive frames. Then, \(p_{r}\) in (21) is given by:  
\[p_{r}=p(r,VF)=\frac{(\lambda VF)^{r}e^{-\lambda VF}}{r!},r=0,1,2,\cdots. \tag{23}\]  
The delay of a packet is defined as the time between this packet's generation and the time when it is cleared from the buffer in terms of time slot. \(G\) is used to represent the event that a generated safety message packet enters the buffer successfully. Let \(T_{j}\) denote the event that the considered packet is generated in the \(j\)_th_ slot within the current \(V\) frames (\(VF\) successive slots). Before the generation of the considered packet, there are total \(i+r\) packets in the buffer under state \(U_{i}\). The considered packet is the \((k+1)\)th packet in the buffer, \(k=i+r-1\). \(X_{k}\) is used to denote this event. According to [41], the average packet delay can be calculated by:  
\[\overline{D}=\sum\limits_{k=0}^{L-1}\sum\limits_{j=1}^{VF}[(VF-j+1)+iVF]P \Big{(}T_{j}X_{k}|G\Big{)} \tag{24}\]  
\[P\Big{(}T_{j}X_{k}|G\Big{)}=\sum\limits_{i=0}^{L-1}\frac{\omega_{i}C_{i}(k,j) }{VFP(G)} \tag{25}\]  
where \(P(G)\) is the probability that a generated packet enters the buffer successfully, and \(C_{i}(k,j)\) is the probability of \(r-1\),\(r=k+1-i\) packet arrivals during the first \(j-1\) slots within the current \(V\) frames.  
\[C_{i}(k,j)=\left\{\begin{aligned} \frac{(\lambda(j-1))^{r-1}e^{- \lambda(j-1)}}{(r-1)!}r<j-1\\ 0,& r\geq j-1\end{aligned}\right. \tag{26}\]  
\[P(G)=\sum\limits_{i=0}^{L-1}\frac{\omega_{i}}{VF}\Bigg{(}\sum\limits_{j=1}^{ L-i}1+\sum\limits_{j=L+1-i}^{VF}\sum\limits_{r=0}^{L-i-1}\frac{(\lambda(j-1))^{r-1 }e^{-\lambda(j-1)}}{r!}\Bigg{)} \tag{27}\]