## 5 Analysis of the Transmission Delay
### One-Hop Distance for the Two-Lane Scenario  
Different from the derivation of the one-hop timer delay, the calculation of distance is based on the lane state transition probability. This is because neither a maximum nor minimum value of \(D^{(1,1)}\) and \(D^{(1,2)}\) can represent \(D^{(1)}\) based on Equation (7). Let \(p_{ij}\) denote the lane state transition probability from the current relay in lane \(l_{i}\) to the next hop relay in lane \(l_{j}\). On the basis of this definition, we can obtain that the CDF of \(D^{(1)}\) as:  
\[\Pr\{D^{(1)}\leq d\}=p_{11}\Pr\{D^{(1,1)}\leq d\}+p_{12}\Pr\{D^{(1,2)}\leq d\}, \tag{32}\]where equations analogous to those given above hold for the CDF random of the variable \(D^{(2)}\), and the derivations of \(p_{ij}\) will be introduced as follows.  
In the two-lane scenario, the transition probability matrix for lane state \(\{X_{n},n>0\}\) contains four items (Figure 4), that is:  
\[P=\left[\begin{array}{cc}p_{11}&p_{12}\\ p_{21}&p_{22}\end{array}\right], \tag{33}\]  
where the transition probability \(p_{11}\) can be calculated out by:  
\[\begin{split} p_{11}&=\Pr\{X_{k+1}=l_{1}|X_{k}=l_{1}\}\\ &=\int\Pr\{T^{(1,1)}=t,T^{(1,2)}>t|T^{(1)}=t\}\,dt\\ &=\int\frac{\Pr\{T^{(1,1)}=t,T^{(1,2)}>t\}}{\Pr\{T^{(1)}=t\}}\,dt\\ &=\int\frac{\Pr\{T^{(1,1)}=t\}(1-\Pr\{T^{(1,2)}\leq t\})}{\Pr\{T^{(1)}=t\}} \,dt.\end{split} \tag{34}\]  
Substituting the results found in Section 5.3, \(p_{11}\) can be represented as:  
\[p_{11}=\int_{t\in I_{3}}\frac{f_{T^{(1,1)}|t\in I_{3}}(t)}{f_{T^{(1)}|t\in I_{ 3}}(t)}\,dt+\int_{t\in I_{4}}\frac{f_{T^{(1,1)}|t\in I_{4}}(t)(1-\Pr\{T^{(1,2) }\leq t|t\in I_{4}\})}{f_{T^{(1)}|t\in I_{4}}(t)}\,dt. \tag{35}\]  
The transition probability can be evaluated by numerical integration of the expression in Equation (35). Specifically, the integration space \(I_{3}\bigcup I_{4}=[t_{1},t_{3}]\) is divided into \(n_{I}\) equally-sized intervals, of length \(e_{t}=(t_{3}-t_{1})/n_{I}\). Then, the integral reduces to a sum. The other three probability items, \(p_{12},p_{21}\) and \(p_{22}\), can be obtained with a similar method.  
Based on the properties of SMP [60], we can obtain the stationary distribution of SMP. It can be deduced that the embedded Markov chain \(\{X_{n},n\geq 0\}\) of the semi-Markov process \(\{Z_{t},t\geq 0\}=\{X_{n},Y_{n},n\geq 0\}\) is irreducible, and the first arrival time is finite. Then, the stationary distribution of \(\{X_{n},n\geq 0\}\) exists, which is denoted as \(\tilde{\pi}=[\tilde{\pi}_{1},\tilde{\pi}_{2}]\). Let \(\pi_{i}\) denote the result of limit \(\lim_{t\rightarrow\infty}\Pr\{Z_{t}=i|Z_{0}=i_{0}\}\), and we have:  
\[\pi_{i}=\frac{\tilde{\pi}_{i}t^{(i)}}{\sum_{j=1}^{2}\tilde{\pi}_{j}t^{(j)}}. \tag{36}\]  
where \(t^{(i)}\) is the mean time spent into state \(i\), before making a transition to the next visited state.  
Based on the lane state stationary distribution probability, we have the expectation of the one-hop transmission distance as:  
\[\mathrm{E}[D_{\mathrm{L}}]=\pi_{1}d^{(1)}+\pi_{2}d^{(2)}. \tag{37}\]  
The difference between the derivations of \(\mathrm{E}[D_{\mathrm{L}}]\) and \(\mathrm{E}[T_{\mathrm{L}}]\) lies in the timer setting rule. We will select the next hop relay with the minimum timer value; however, this relay vehicle might not be the furthest vehicle among the current relay's neighbors. Therefore, we take the method proposed in Equation (37) as an alternative to obtain the expectation of \(D_{\mathrm{L}}\).  
Figure 4: Two-lane state Markov chain transitions.