# 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 one-step transition probabilities are:  
\[\left\{\begin{aligned} & P\left\{i,k|i,k\right\}=P_{s,b}\ \ \ \ 0\leq k\leq W_{s,i}-1,0\leq  
According to the stationary distribution of the Markov chain, we express the total probability as:  
\[b_{I_{e}}+\sum_{k=0}^{W_{e}-1}b_{e,k}=1 \tag{26}\]  
\[b_{I_{s}}+\sum_{i=0}^{m}\sum_{k=0}^{W_{s,i}-1}b_{s,i,k}=1 \tag{27}\]  
Using Equations (24) and (26), internal transmission probability \(\tau_{e}\) of a safety-related packet in a randomly slot time is  
\[\tau_{e}=b_{e,0}=\left[\frac{W_{e}-1}{2(1-P_{e,b})}+\frac{P_{e,emp}+P_{e,arr}}{ P_{e,arr}}\right]^{-1} \tag{28}\]  
Combining Equations (25) and (27), we get  
\[b_{s,0,0}=\left\{ \begin{array}{l}\left[\frac{W_{s,0}\left[1-\left(2P_{s,f} \right)^{m+1}\right]}{2(1-P_{s,b})(1-2P_{s,f})}-\frac{1-P_{s,f}^{m+1}}{2(1-P_{ s,b})(1-P_{s,f})}\right.\\ \left.\hskip 14.226378 pt+\frac{P_{s,emp}}{P_{s,arr}}+\frac{1-P_{s,f}^{m+1}}{1-P _{s,f}}\right]^{-1},\quad m\leq m^{\prime}\\ \\ \left[\frac{W_{s,0}\left[1-\left(2P_{s,f}\right)^{m^{\prime}+1}\right]}{2(1-P _{s,b})(1-2P_{s,f})}\right.\\ \left.\hskip 14.226378 pt+\frac{2^{m^{\prime}}W_{s,0}(P_{s,f}^{m^{\prime}+1}-P_{ s,f}^{m+1})+P_{s,f}^{m+1}-1}{2(1-P_{s,b})(1-P_{s,f})(1-P_{s,f})}\right.\\ \left.\hskip 14.226378 pt+\frac{P_{s,emp}}{P_{s,arr}}+\frac{1-P_{s,f}^{m+1}}{1-P _{s,f}}\right]^{-1},\quad m>m^{\prime}\end{array} \right. \tag{29}\]  
As any transmission occurs when the backoff timer is equal to zero, regardless of the backoff stage, the internal transmission probability \(\tau_{s}\) that a node transmits a WSA packet in a random time slot can be given by  
\[\tau_{s}=\sum_{i=0}^{m}b_{s,i,0}=\sum_{i=0}^{m}p_{s,f}^{i}b_{s,0,0}=\frac{1-P_ {s,f}^{m+1}}{1-P_{s,f}}b_{s,0,0} \tag{30}\]  
Now, we calculate **average virtual slot \(T_{virt}\)**[40]. Let \(T_{e,suc}\) (\(T_{e,col}\)) and \(T_{s,suc}\) (\(T_{s,col}\)) denote the time of successful transmission (transmission collision) of a safety packet and the time of successful reservation (transmission collision) of a WSA packet, respectively. Due to our Assumption 4 in Section 4.1, the time for transmission collision equals to the time for transmission bit error, and thus, for simplification, \(T_{e,col}\) (\(T_{s,col}\)) also denotes the time for transmission bit error of safety (WSA) packet. We have  
\[T_{e,suc}=T_{e,col}=T_{e}+\text{AIFS}[0] \tag{31}\]  
\[\left\{ \begin{array}{l}T_{s,suc}=T_{s}+\text{AIFS}[1]+SIFS+T_{ACK}+\delta\\ T_{s,col}=T_{s}+\text{AIFS}[1]\end{array} \right. \tag{32}\]  
where \(L_{ACK}\) and \(T_{ACK}=E[L_{ACK}]/R_{d}\) denote the packet size of an ACK packet and the time for transmitting an ACK packet, respectively. Let \(P_{e,suc}^{p}\) and \(P_{s,suc}^{p}\) be the probability that the transmission attempt of an safety packet and a WSA packet are successful, respectively, which have no collisions caused by: the transmissions of higher ACs within the same station, concurrent transmissions of the other stations within the sensing-carrier range, the hidden terminal problem and the packets dropped due to more than retransmission limit for WSA packet, conditioned on the fact that at least one station transmits in the considered time slot. By definition,\[\left\{