# Performance Analysis of the IEEE 802.11p Multichannel MAC Protocol in Vehicular Ad Hoc Networks
## 4 Analytical Model and Performance Analysis
### Markov Chain for Safety-Related Message and WSA Message
The channel sensed busy means that there is at least one node transmitting any type of packets in the carrier-sensing range \(L_{cs}\). Thus, \(P_{busy}\) can be calculated by  
\[P_{busy}=1-\sum_{k=0}^{\infty}{(1-\eta_{total})^{k}\frac{N_{cs}}{k!}e^{-N_{cs} }}=1-e^{-\eta_{total}\cdot N_{cs}} \tag{20}\]  
We analyze both one-dimensional and two-dimensional Markov chains to find the one-step transition probabilities. Let \(b_{e}(t)\) denote a stochastic process representing the backoff timer value of safety traffic at time \(t\). Since the transmission mode of safety packets is broadcast without backoff stages, so the fixed CW is denoted by \(W_{e}\). The state of safety packet is described by \(\{k;k\in[0,W_{e}-1]\}\). \(I_{e,idle}\) stands for an idle state of an EDCA node with an empty safety queue. The stationary distribution of the idle state and the backoff state \(k\) are denoted by \(b_{I_{e}}\) and \(b_{e,k}\), respectively. According to Figure 7a, the one-step transition probabilities are:  
\[\left\{ \begin{array}{ll}P\left\{k|k+1\right\}=1-P_{e,b}&0\leq k\leq W_{e}-2 \\ P\left\{k|0\right\}=\frac{1-P_{e,emp}}{W_{e}}&0\leq k\leq W_{e}-1\\ P\left\{k|k\right\}=P_{e,b}&0\leq k\ \ \leq W_{e}-1\\ P\left\{L_{e,idle}|0\right\}=P_{e,emp}\\ P\left\{L_{e,idle}|L_{e,idle}\right\}=1-P_{e,arr}\\ P\left\{k|L_{e,idle}\right\}=\frac{P_{e,arr}}{W_{e}}&0\leq k\leq W_{e}-1\end{array} \right. \tag{21}\]  
The meaning of each line in Equation (21) is as follows:  
1. If the channel is idle, the backoff timer subtracts one. 2. If the backoff timer value is zero and the queue is not empty, then the backoff timer is initially uniformly chosen in \([0,W_{e}-1]\). 3. If the channel is sensed busy, the backoff timer is frozen. 4. If the backoff timer value is zero and the queue is empty, the node is under idle state. * If no packets are to be sent in the queue and no generated packet arrives at the queue, then the node stays in the idle state. * If the queue is empty, a new arriving packet makes the backoff timer uniformly chosen in \([0,W_{e}-1]\). The backoff process of WSA traffic with AC\({}_{1}\) is modeled as a two-dimensional Markov chain with discrete-time evolved from [7; 18; 29; 37; 41; 42; 43], as shown in Figure 7b. Let \(s(t)\) and \(b(t)\) be a stochastic process standing for the backoff stage and backoff timer of WSA traffic, respectively. Let the state \(I_{s,idle}\) represent that the AC\({}_{1}\) queue of an EDCA node is empty. The maximum retransmission number and the CW in the \(i\)th backoff stage are denoted by \(m\) and \(W_{s,i}\) (\(i\in[0,m]\)), respectively. CW is set to the minimum value \(W_{s,0}\) for the first transmission attempt. When collision is detected, CW is doubled and then retransmission is started for the first \(m^{\prime}\) steps. While the CW remains unchanged for \((m-m^{\prime})\) steps. To sum up, if \(W_{s,i}\) is the CW size in the \(i\)th step:  
\[W_{s,i}\ =\ \left\{ \begin{array}{ll}2^{i}W_{s,0}&\qquad i\leq\ \ m^{\prime}\\ 2^{m^{\prime}}W_{s,0}&\qquad m^{\prime}<\ \ i\ \leq m\end{array} \right. \tag{22}\]  
Figure 7: Markov chain backoff procedure: (**a**) one-dimensional Markov chain model for higher priority AC\({}_{0}\); and (**b**) two-dimensional Markov chain model for lower priority AC\({}_{1}\). Let \(b_{s,i,k}=\lim_{t\rightarrow\infty}\left\{s(t)=i,b(t)=k\right\},0\leq i\leq m,0\leq k \leq W_{s,j}-1\) be the stationary distribution of the chain of WSA traffic in Figure 7b. The stationary distribution of the idle state and the backoff state are denoted by \(b_{l_{s}}\) and \(b_{s,i,k}\), respectively.