# Performance Analysis of the IEEE 802.11p Multichannel MAC Protocol in Vehicular Ad Hoc Networks
## 4 Analytical Model and Performance Analysis
### Performance Analysis  
#### 4.4.1 Packet Transmission Delay  
**Definition 4** (_Packet transmission delay_). : _The packet transmission delay is the time duration between the time instant that a packet arrives at the queue and the time instant that this packet is successful transmitted or dropped. There are two cases for packet dropped: on case is that for a safety packet, the packet is dropped due to expiration. The other case is that a WSA packet is dropped due to reaching the maximum retransmission number._  
Due to the channel switching, each packet arriving at the MAC queue during SCHI has to wait, until the end of SCHI to be transmitted. Since the time of packets arriving at the MAC queue is random, the delay in the SCHI can be approximated by half of SCH interval. On the other hand, when a node contends for CCH (a common server) to transmit a packet during the CCHI, the average transmission delay of an \(i\) (\(i\in[e,s]\)) packet in the CCHI consists of service time \(TS_{i}\) and queuing time \(TQ_{i}\). Let \(D_{e}\), \(D_{s}\) and \(T_{SCHI}\) denote the total transmission delay of a safety packet, the total transmission delay of a WSA packet and the duration of SCH interval, respectively, we have  
\[\left\{ \begin{aligned} D_{e}&=\frac{T_{ SCHI}}{2}+TS_{e}+TQ_{e}\\ D_{s}&=\frac{T_{SCHI}}{2}+TS_{s}+TQ_{s}\end{aligned} \right. \tag{38}\]  
Now, we calculate \(TS_{e}\) (\(TS_{s}\)) and \(TQ_{e}\) (\(TQ_{s}\)). **Definition 5** (_Service time_). : _The service time \(TS_{i}\) is the duration from the time instant when an \(i\) (\(i\in[e,s]\)) packet becomes the head of the queue to the time instant when this packet is transmitted or dropped. The service time is important to examine the performance of higher protocol layer [39]. Since the smallest time unit of backoff timer is a time slot \(\sigma\), the service time of packets is a nonnegative random variable whose distribution is a discrete probability distribution. In this paper, by following the similar approach in [39; 45], we model each entity as an M/G/1 queue with unlimited length and apply the Probability Generating Function (PGF) approach to transform the Markov chain into the \(z\) domain, as shown in Figure 8. Let \(q_{i,k}\) be the steady state probability of \(i\)\((i\in[e,s])\) packets that the packet service time is \(\text{ts}_{i,k}\cdot\sigma\). Let \(P_{TS_{i}}\) denote the PGF of \(q_{i,k}\), which is expressed as  
\[P_{TS_{i}}(z)=\sum_{k=0}^{\infty}q_{i,k}z^{\text{ts}_{i,k}} \tag{39}\]  
The service time includes the backoff time and the transmission time. In the EDCA backoff procedure, for a safety packet, the backoff timer will be decremented by a time slot (\(\sigma\)) with probability \((1-P_{e,b})\), while it is frozen for a period of time when the channel is busy. There are five cases making backoff timer frozen: safety packet error with probability \(P^{2}\), successful transmission a safety packet with probability \(P^{3}\), WSA packet error with probability \(P^{4}\), successful transmission a WSA packet with probability \(P^{5}\), collision slots from only emergency packets, only WSA packets or both with probability \(P_{e,b}-P^{2}-P^{3}-P^{4}-P^{5}\). Let \(S_{e}(z)\) and \(C_{e}(z)\) be the PGF of \(T_{e,suc}\) and \(T_{e,col}\), respectively, we have  
\[S_{e}(z)=C_{e}(z)=z^{\left\lfloor\frac{T_{e,suc}}{\sigma}\right\rfloor} \tag{40}\]  
Figure 8: Generalized state transition diagram: (**a**) safety-related packets with higher priority \(\text{AC}_{0}\); and (**b**) WSA packets with lower priority \(\text{AC}_{1}\). where \(\lfloor\cdot\rfloor\) is a function to round floating point numbers to integers. Therefore, the PGF of the time that the backoff timer of safety packet decrements by one can be expressed by  
\[ \begin{split} H_{e}(z)&=(1-P_{e,b})z+P^{2}C_{e}(z)+P^{3 }S_{e}(z)+P^{4}C_{s}(z)\\ &\quad+P^{5}S_{s}(z)+(P_{e,b}-\sum_{i=2}^{5}P^{i})\cdot\max[C_{e} (z),C_{s}(z)]\end{split}