# Modeling and Performance of the IEEE 802.11p Broadcasting for Intra-Platoon Communication
## 4 Analytical Model
### MAC Layer Queuing Process Model  
According to [34], the control message generation rate is dynamically adjusted. It is related to the road condition, vehicle behavior, and traffic condition. If there is an increase in the vehicle movement dynamics, the packets arrival rate increase to ensure the movement dynamic is correctly reflected in the message. Thus, the transmission of messages generated by a vehicle in a platoon can be regarded as a random event in practice. We assume the probability of packets arrival conforms to the Poisson distribution with a parameter of \(\lambda\), where \(\lambda\) represents the number of packets arriving in a unit time. When a vehicle has packets to transmit, these packets will be sent to the queue buffer. Packets wait in the buffer according to the principle of first in first out. Certain packet access the communication resources when it reaches to the head of the queue. M/G/1/K queuing theory is applied to simulate the queuing process, and the maximum length of a queue is \(K\). The time from channel competition to successful transmission or unsuccessful transmission can be defined as the packet service time [35], which generally set as integer times of the idle slot. We define the service time distribution as \(p_{st}^{T}\), \(T\in[0,T_{m}]\). The maximum service time is \(T_{m}\tau\), where \(\tau\) is the free slot. Following [36], we define \(X(t),(t\geq 0)\) as the state of the queuing system at time \(t\). Let \(\delta_{j}\) be the time instance of the \(j^{th}\) packet departure. We consider the embedded Markov process \(\{X_{j}\}\), where \(X_{j}\) is the state of the queuing system just before \(\delta_{j}\). The state space of the embedded Markov process can be represented as  
\[S^{\prime}=\{A_{0},A_{1},A_{2},\cdots,A_{K}\} \tag{11}\]  
It should be noted that \(A_{K}\) indicates \(K-1\) packets wait in the queue and one is serviced. When the size of buffer \(K\) is set to be one, the state space of the embedded Markov process only contains two states: \(S^{\prime}=\{A_{0},A_{1}\}\). \(A_{0}\) is the state that the queue is emptied and no packet is serviced and \(A_{1}\) represents the state that no packets wait in the queue and one packet is serviced [31]. In this condition, a new incoming packet is immediately served without waiting. Define \(a_{k}\) as the probability that \(k\) packets arrive during one packet service time. As the packet arrival is a Poisson process with rate \(\lambda\), we have  
\[a_{k}=\sum_{T=0}^{T_{m}}p_{st}^{T}\frac{(\lambda T\tau)^{k}}{k!}e^{-\lambda T\tau} \tag{12}\]  
We suppose \(p_{st}^{T}=0\) represents no packets arrival. Then, the probability that the sending buffer of a communication vehicle has new packets arrival in a slot time equals  
\[p_{na}=1-e^{-\lambda T_{idle}} \tag{13}\]  
where \(T_{idle}\) is the average idle time of the vehicle buffer when the channel is busy. \[ \begin{split} T_{idle}&=\tau[1-(1-q_{em})p]^{N-1}+T _{c}\{[(1-q_{em})p]^{N-1}\\ &-(N-1)(1-q_{em})p[1-(1-q_{em})p]^{N-2}\}\\ &+T_{e}p_{e}(N-1)(1-q_{em})p[1-(1-q_{em})p]^{N-2}\\ &+T_{s}(1-p_{e})(N-1)(1-q_{em})p[1-(1-q_{em})p]^{N-2}\end{split}  \tag{14}\]In Equation (14), \(T_{s}\) and \(T_{e}\) are the length of time of successful and collided packet transmission, respectively. \(T_{s}\) and \(T_{e}\) can be expressed as follows  
\[T_{e}=T_{s}=DIFS+H_{phy}+H_{mac}+D_{l} \tag{15}\]  
In addition, \(T_{c}\) is the time consumed by the channel conflict in the process of data transmission and can be calculated as  
\[T_{c}=DIFS \tag{16}\]  
Let \(p_{k}^{que}(0\leq k\leq K)\) be the steady-state probability that \(X_{j}=A_{k}\), which is  
\[p_{k}^{que}=\left\{ \begin{array}{l}p_{0}^{que}a_{k}+\sum_{i=1}^{k+1}p_{i}^{que }a_{k-i+1},\;k=0,1,\cdots,K-1\\ p_{0}^{que}(1-\sum_{l=0}^{k-1}a_{l})+\sum_{i=1}^{k}p_{i}^{que}(1-\sum_{l=i-1}^ {k-1}a_{l-i+1}),\;k=K\end{array} \right.