# Modeling and Performance of the IEEE 802.11p Broadcasting for Intra-Platoon Communication
## 4 Analytical Model  
### Intra-Platoon Back-Off Process  
In this section, we analyze the performance of IEEE 802.11p broadcast for intra-platoon communication in a single platoon scenario. In IEEE 802.11p broadcast mode, vehicles with fresh information to be transmitted monitor the channel states and use back-off mechanism to compete for channels. Before the back-off process, one vehicle needs to choose a random number within \([0,CW]\) as the back-off time. Once the initial contention window \(CW\) is defined, it will keep constant for all back-off processed because there is no retransmission in the broadcast mode. The state of back-off time counter can be classified into three categories: (1) The back-off time counter is reduced by one as long as the channel is detected idle; (2) The back-off counter is frozen when a packet is transmitted on the channel; (3) If the channel is detected idle again over a DIFS, the back-off counter is reactivated. When a vehicle's backoff-time counter reaches zero, the vehicle instantly broadcasts the information.  
In the analysis, we make the following hypotheses: ideal channel conditions (i.e., no hidden terminals and capture); vehicles do not have packets available for transmission at all time, which indicates the data traffic condition is unsaturated; and there are channel impairments under a practical channel condition, so a packet can encounter transmission errors.  
We assume that the bit error rate is \(p_{b}\). Then, \(p_{e}\) the transmission error probability of packers caused by channel error can be expressed as  
\[p_{e}=1-(1-p_{b})^{L} \tag{6}\]  
and \(L=H_{phy}+H_{mac}+D_{l}\) is the valid data length. \(D_{l}\) denotes data size, \(H_{phy}\) represents the size of data in PHY and \(H_{mac}\) represents the size of data in MAC. Packets sent by a vehicle can be received by all vehicles in the same platoon due to vehicles in a platoon are within the communication range of each other. Thus, the probability of channel congestion can be computed as  
\[p_{p}=1-[1-(1-q_{em})p]^{N-1} \tag{7}\]  
where \(N\) represents the platoon size (vehicles per platoon), and \(q_{em}\) denotes new packets arrival probability. According to \(p_{p}\) and \(p_{e}\), the probability of unsuccessful transmission can be expressed as  
\[p_{f}=1-(1-p_{p})(1-p_{e})=p_{p}+p_{e}-p_{p}p_{e} \tag{8}\]  
We define \(p_{na}\) as the arrival rate of a new packet, and \(q_{em}\) as the probability that a vehicle does not send data packets in a slot time. Then, the probability that a channel is detected as idle in the back-off process can be represented as \(q_{em}^{N}\). When a vehicle's sending queue is not empty, no matter how many packets there are in the sending buffer, the vehicle will contend for the channel in the same way according to the IEEE 802.11p broadcast. Stochastic process \(b(t)\) and \(s(t)\) are defined as the back-off time counter and the back-off order for a certain vehicle, respectively. Then, \(\{s(t),b(t)\}\) is modeled with a 1-D Markov-chain. Let \(b_{r}=\lim_{t\rightarrow\infty}P\{b(t)=r\}\), \(0\leq r\leq CW\) represent the stationary distribution, and \(b_{N}\) be the empty state of buffer. According to the Markov-chain model, we have  
\[\left\{\begin{array}{l}b_{r}=\frac{(b_{0}+b_{N})p_{na}}{(CW+1)q_{em}^{N}}+b _{r+1},\ r=0,1,\cdots,CW-1\\ b_{r}=\frac{(b_{0}+b_{N})p_{na}}{(CW+1)q_{em}^{N}},\ r=CW\\ \sum_{k=0}^{CW}b_{r}+b_{N}=1\end{array}\right. \tag{9}\]  
Solving Equation (9), the transmission attempt probability of packets in unsaturated flow condition is given as follow\[p=b_{0}=\frac{2q_{em}^{N}p_{na}}{CWp_{na}+2q_{em}^{N}} \tag{10}\]