## 5 Analysis of the Transmission Delay
### One-Hop Timer Delay for the Two Lane Scenario  
Since \(I_{1}=[t^{-}_{1,1},t^{+}_{1,1}]\) and \(I_{2}=[t^{-}_{1,2},t^{+}_{1,2}]\), where the limits are determined by parameter \(\epsilon\), a special case of \(\epsilon\) is taken as an example. Suppose that \(\epsilon=1/2\), \(I_{1}\) and \(I_{2}\) can be divided into two sub-intervals, \(I_{3}\triangleq[t_{1},t_{2}]\) and \(I_{4}\triangleq[t_{2},t_{3}]\), where \(t_{1}=t_{\min}\), \(t_{2}=t_{\min}+t_{\Delta}(1-\epsilon/(2v_{\Delta}L_{\max}))\) and \(t_{3}=t_{\min}+t_{\Delta}/2\). Let \(T^{(1)}\) denote the minimum timer delay conditioned on the sender vehicle being located in lane \(l_{1}\).  
The derivation of \(T^{(1)}\) will be discussed in two cases as follows. When \(t\in I_{3}\), we have:  
\[\Pr\{T^{(1)}\leq t|t\in I_{3}\}=\Pr\{T^{(1,1)}\leq t|N>0\}=\frac{1-\exp\left(- \lambda_{1}\left(\alpha_{1}t+\beta_{1}\right)\right)}{1-\exp(-\lambda_{1})}. \tag{28}\]  
Otherwise, when \(t\in I_{4}\), we have:  
\[\begin{split}\Pr\{T^{(1)}\leq t|t\in I_{4}\}=& 1-\Pr\{T^{(1)}>t|t\in I_{4}\}\\ =& 1-\Pr\{T^{(1,1)}>t\}\cdot\Pr\{T^{(1,2)}>t\}\\ =& 1-(1-\Pr\{T^{(1,1)}\leq t\})\cdot(1-\Pr\{T^{(1,2)} \leq t\}),\end{split} \tag{29}\]  
where we have not included the condition \(N>0\) to simplify the expression of the probabilities.  
Substituting the CDF of \(T^{(1,1)}\) and \(T^{(1,2)}\) into the above equations and taking the derivate, we can obtain the pdf of \(T^{(1)}\) in case \(t\in I_{4}\) as:  
\[\begin{split} f_{T^{(1)}|t\in I_{4}}(t)=&\frac{ \lambda_{1}\alpha_{1}\exp\left(-\lambda_{1}\left(\alpha_{1}t+\beta_{1}\right) \right)}{1-\exp(-\lambda_{1})}+\frac{\lambda_{2}\alpha_{2}\exp\left(-\lambda_ {2}\left(\alpha_{2}t+\beta_{2}\right)\right)}{1-\exp(-\lambda_{2})}\\ -&\frac{\lambda_{1}\alpha_{1}\exp\left(-\lambda_{1} \left(\alpha_{1}t+\beta_{1}\right)\right)+\lambda_{2}\alpha_{2}\exp\left(- \lambda_{2}\left(\alpha_{2}t+\beta_{2}\right)\right)}{[1-\exp(-\lambda_{1})] [1-\exp(-\lambda_{2})]}\\ +&\frac{\left(\lambda_{1}\alpha_{1}+\lambda_{2} \alpha_{2}\right)\exp\left(-\left(\left(\lambda_{1}\alpha_{1}+\lambda_{2} \alpha_{2}\right)t+\left(\beta_{1}+\beta_{2}\right)\right)\right)}{[1-\exp(- \lambda_{1})][1-\exp(-\lambda_{2})]}.\end{split} \tag{30}\]  
Let \(t^{(1)}\) denote the expectation of \(T^{(1)}\). Then, we have:  
\[\begin{split} t^{(1)}=&\int_{t\in I_{3}}t\cdot f_{T^ {(1)}|t\in I_{3}}(t)\,dt+\int_{t\in I_{4}}t\cdot f_{T^{(1)}|t\in I_{4}}(t)\,dt \\ =&-\frac{1}{\mathcal{C}_{1}}\left(t+\frac{1}{\lambda_ {1}\alpha_{1}}\right)\exp\left(-\lambda_{1}\left(\alpha_{1}t+\beta_{1}\right) \right)|_{I_{1}}^{t_{3}}\\ &+\left(\frac{1}{\mathcal{C}_{1}\mathcal{C}_{2}}-\frac{1}{ \mathcal{C}_{2}}\right)\left(t+\frac{1}{\lambda_{2}\alpha_{2}}\right)\exp \left(-\lambda_{2}\left(\alpha_{2}t+\beta_{2}\right)\right)|_{I_{2}}^{t_{3}}\\ &+\frac{1}{\mathcal{C}_{1}\mathcal{C}_{2}}\left(t+\frac{1}{ \lambda_{1}\alpha_{1}}\right)\exp\left(-\lambda_{1}\left(\alpha_{1}t+\beta_{1 }\right)\right)|_{I_{2}}^{t_{3}}\\ &-\frac{1}{\mathcal{C}_{1}\mathcal{C}_{2}}\left(t+\frac{1}{ \left(\lambda_{1}\alpha_{1}+\lambda_{2}\alpha_{2}\right)}\right)\exp\left(- \left(\left(\lambda_{1}\alpha_{1}+\lambda_{2}\alpha_{2}\right)t+\left(\beta_{1 }+\beta_{2}\right)\right)\right)|_{I_{2}}^{t_{3}}.\end{split} \tag{31}\]  
Based on the aforementioned definition, the obtained value of \(t^{(1)}\) represents \(E[T_{\text{L}}]\) under the condition that the sending vehicle node is located in lane \(l_{1}\) and that there is at least one vehicle in the transmission range of the sending node, i.e., \(N>0\). In a similar way, we can obtain the probability density function and the mean value of the random variable \(T^{(2)}\), defined as the minimum timer delay conditioned on the sender vehicle being located in lane \(l_{2}\).