# 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  
In our analytical model, each EDCA node can deliver two kinds of messages on the CCH: AC\({}_{0}\) and AC\({}_{1}\) corresponding to safety-related message and WSA message, respectively. Due to the broadcast mechanism of the transmission of the safety-related message without ACK, the sender cannot detect the external collision, and thus no retransmission will be trigged by the external collision. Therefore, the backoff procedure of safety-related message is modeled as a one-dimensional Markov chain, as shown in Figure 7a. On the other hand, the nodes use a pair of control messages such as WSA and ACK to perform negotiations on the CCH, and retransmission will be trigged due to the missing ACK. Thus, a two-dimensional Markov chain model is utilized to analyze the backoff procedure of WSA message, as shown in Figure 7b. We use Figure 7a,b to obtain the stationary internal probability \(\tau_{e}\) and \(\tau_{s}\), respectively. In our analytical model, unsuccessful transmission happens in two cases: (1) transmission collision due to virtual (internal) collision with the higher AC inside the same station or due to external collision with other stations; and (2) transmission error due to the unfavorable channel fading. We assume that the transmission collision or transmission error is constant and independent of the number of collisions or the number of errors of this packet has suffered in the past. Probabilities used in two Markov chain models are listed as follows:  
\(\mathbf{P_{i,arr}}\): The arrival probability of \(i(i\in[e,s])\) packets. We assume that the generated packets arrive at the MAC queues follow Poisson distributions, and the probability of receiving at least one packet in a virtual slot \(T_{virt}\) is:  
\[P_{i,arr}=1-e^{-2\lambda_{i}T_{virt}} \tag{14}\]  
\(\mathbf{P_{i,emp}}\): The probability of \(i(i\in[e,s])\) queue being empty. \(P_{i,emp}\) can be calculated by:  
\[P_{i,emp}=1-\rho_{i} \tag{15}\]  
where \(\rho_{i}\) is the probability that at least one packet stays in \(i(i\in[e,s])\) queue, and it also stands for the busy probability of server for \(i\) queue. \(\rho_{i}\) can be derived by:  
\[\rho_{i}=\frac{2\lambda_{i}}{\sum\limits_{i\in[e,s]}\mu_{i}} \tag{16}\]  
where \(\mu_{i}\) is the average service rate of virtual \(i(i\in[e,s])\) queue (in terms of packets per second). \(\mathbf{P_{i,b}}\): The backoff blocking probability (i.e., frame blocking probability defined in [18]). If a node senses the channel is busy due to other nodes or the other AC queues in the same node occupying the channel, the backoff timer is frozen (suspended). We have [37]  
Figure 6: Hidden terminal vulnerable period. \[P_{i,b} =1-\left[\sum_{k=0}^{\infty}{(1-\eta_{total})^{k}\frac{(N_{cs}-1)^{k }}{k!}e^{-(N_{\alpha}-1)}}\right. \tag{17}\] \[\quad\cdot\prod_{ \begin{subarray}{c}j\in[e,s]\\ j\
eq i\end{subarray} }{(1-\tau_{j})}\Bigg{]}^{\text{AIFSN}[[j]-\text{AIFSN}[0] +1}\] \[=1-\left[e^{-(N_{\alpha}-1)\eta_{total}}\cdot\prod_{ \begin{subarray}{c}j\in[e,s]\\ j\
eq i\end{subarray} }{(1-\tau_{j})}\right]^{\text{AIFSN}[[j]-\text{AIFSN}[0] +1}\]  
Note, in term AIFSN[\(i\)], \(i=e\) denotes AIFSN[0], and \(i=s\) denotes AIFSN[1]. \(\mathbf{P}_{i,f}\): The transmission failure probability of an \(i(i\in[e,s])\) packet. A successful packet transmission occurs only if there no collision and no bit error during the transmission attempt. Therefore, \(P_{i,f}\) can be derived as [32,41]:  
\[P_{i,f}=1-(1-P_{i,e})(1-P_{i,err}) \tag{18}\]  
Note that the interpretation of \(P_{i,f}\) (\(i\in[e,s]\)) is different from \(p\) in [29], as it also accounts for the possibility of unsuccessful transmission caused by bit errors. \(\mathbf{P}_{s,drop}\): The probability that a WSA packet is dropped due to more than retransmission limit \(m\). It can be calculated by  
\[P_{s,drop}=P_{s,f}^{m+1} \tag{19}\]  
\(\mathbf{P}_{busy}\): The channel busy probability.