# 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
\begin{array}{l}P_{e,suc}^{v}=\dfrac{N_{tr}\times\eta_{e}\times\sum \limits_{k=0}^{\infty}{(1-\eta_{total})^{k}\frac{(N_{cs}-1)^{k}}{k!}e^{-(N_{cs}- 1)}}}{P_{busy}}\\ \qquad\qquad\times\left[\sum\limits_{k=0}^{\infty}{(1-\eta_{total})^{k}\frac{ N_{\rm int}^{k}}{k!}e^{-N_{\rm int}}}\right]^{\mathbb{T}_{valin}/T_{virt}}\\ \qquad\qquad=\dfrac{N_{tr}\times\eta_{e}\times e^{-\eta_{total}(N_{cs}+N_{\rm int }T_{valin}/T_{virt}-1)}}{P_{busy}}\\ P_{s,suc}^{v}=\dfrac{N_{tr}\times\eta_{s}\times\sum\limits_{k=0}^{\infty}{(1- \eta_{total})^{k}\frac{(N_{cs}-1)^{k}}{k!}e^{-(N_{cs}-1)}}}{P_{busy}}\\ \qquad\qquad\times\left[\sum\limits_{k=0}^{\infty}{(1-\eta_{total})^{k}\frac{ N_{\rm int}^{k}}{k!}e^{-N_{\rm int}}}\right]^{T_{valin}/T_{virt}}\\ \qquad\qquad\qquad\times(1-P_{s,drop})\\ \qquad\qquad=\dfrac{N_{tr}\times\eta_{s}\times e^{-\eta_{total}(N_{cs}+N_{\rm int }T_{valin}/T_{virt}-1)}}{P_{busy}}\\ \qquad\qquad\qquad\times(1-P_{s,drop})\\ \qquad\qquad\qquad\qquad\times(1-P_{s,drop})\end{array} \right. \tag{33}\]  
Let \(P_{col}\) be the probability that a transmission fails due to a collision given that there is at least one other station transmits in the considered time slot, we have [37]  
\[P_{col}=P_{busy}-P_{busy}\sum\limits_{i\in[e,s]}P_{i,suc}^{v} \tag{34}\]  
Based on our Assumption 4 in Section 4.1, there are six types of virtual slots : empty slots (all nodes are in backoff procedure or idle), safety packet error slots, successful transmitting safety packet slots, WSA packet error slots, successful transmitting WSA packet slots, collision slots from only emergency packets, only WSA packets or both, and let \(T^{1}\), \(T^{2}\), \(T^{3}\), \(T^{4}\), \(T^{5}\) and \(T^{6}\) be the duration, let \(P^{1}\), \(P^{2}\), \(P^{3}\), \(P^{4}\), \(P^{5}\), and \(P^{6}\) be the probabilities, corresponding to the above slots, respectively. Their durations and probabilities are expressed by Equations (35) and (36), respectively. \[\left\{ \begin{array}{l}T^{1}=\sigma\\ T^{2}=T_{e,col}\\ T^{3}=T_{e,suc}\\ T^{4}=T_{s,col}\\ T^{5}=T_{s,suc}\\ T^{6}=\max(T_{e,col},T_{s,col})\end{array} \right. \tag{35}\]  
\[\left\{ \begin{array}{l}P^{1}=1-P_{busy}\\ P^{2}=P_{busy}P_{e,suc}^{v}\left[1-(1-p_{ber})^{E[L_{c}]}\right]\\ P^{3}=P_{busy}P_{e,suc}^{v}(1-p_{ber})^{E[L_{c}]}\\ P^{4}=P_{busy}P_{s,suc}^{v}\left[1-(1-p_{ber})^{E[L_{s}]}\right]\\ P^{5}=P_{busy}P_{s,suc}^{v}(1-p_{ber})^{E[L_{s}]}\\ P^{6}=P_{col}\end{array} \right. \tag{36}\]Therefore, \(T_{virt}\) can be expressed as  
\[T_{virt}=\sum_{i=1}^{6}P^{i}T^{i} \tag{37}\]  
Using Equations (2)-(15), (17)-(20), (22) and (28)-(37), variables \(\tau_{e}\), \(\tau_{s}\), \(P_{e,b}\), \(P_{s,b}\), \(P_{e,f}\) and \(P_{s,f}\) can be solved by the numerical methods as in [29], where \(0<\tau_{e},\tau_{s},P_{e,b},P_{s,b},P_{e,f},P_{s,f}<1\). Our Markov chain models are derived from the fundamental developments of the models by [PERSON] [29], and extend the [PERSON]'s models to prioritized scheme by introducing multiple ACs with distinct parameters setting. Moreover, we also take into account virtual collisions, the retry limit, channel switch, channel fading, unsaturated conditions, and the difference among frame collision probability, frame failure probability, frame blocking probability, and channel busy probability. Note that if \(P_{e,emp}=1-\rho_{e}=1\) or \(P_{s,emp}=1-\rho_{s}=1\), it means that there is no packet to transmit and it is the extreme unsaturated condition. This is similar to the Markov chain model proposed in [17; 18] under bit error free conditions. On the other hand, if \(P_{e,emp}=1-\rho_{e}=0\) or \(P_{s,emp}=1-\rho_{s}=0\), it implies that there is always packet to transmit and it is the saturated condition. This is similar to the Markov chain model proposed in [37] under saturation and bit error free conditions. Thus, our model covers the extreme unsaturated, intermediate unsaturated, and saturated conditions by taking the queuing status (\(P_{e,emp}\), \(P_{s,emp}\)) into account. The value of \(P_{e,emp}\) and \(P_{s,emp}\) will be derived later in the next section.