## 5 Analysis of the Transmission Delay
### Preliminary  
To describe the signal power attenuation, we adopt the path-loss model derived from highway measurements in [54]. Hence, given the sensitivity of the OBU receiver, the radio range is a constant, denoted as \(R\). The vehicular traffic is assumed to be distributed along the road according to a one-dimensional Poisson point process, which has been observed in many previous works [53; 59]. For the tractability of the ensuing analysis, the vehicular speed is modeled discretely, where the number of the discrete speed values is in accordance with the number of lanes. That is, in the rest of the analysis, we assume that vehicles in a specific lane keep a given speed. As shown in Figure 2, a two lane highway scenario is adopted to show the theoretical methodology. Let \(\rho_{i}\) denote the traffic density (vehicles per meter) and \(v_{i}\) denote the expected speed (meters per second) for the \(i\)-th road lane. Then, the average number of vehicles in the radio range \(R\) can be represented as \(\lambda_{i}\), where \(\lambda_{i}=\rho_{i}R\) and \(i\in\{1,2\}\).  
We adopt the Semi-Markov Process (SMP) for analyzing the delay issue. A continuous-time stochastic process is an SMP when the embedded state transition chain (the discrete states representing in which lane the relay vehicle is located) is a Markov chain, and the transition times are random variables, whose probability distribution function (pdf) depends on the two states between which the transition is made. The SMP can be characterized by:  
\[X_{0}\xrightarrow{Y_{0}}X_{1}\xrightarrow{Y_{1}}X_{2}\xrightarrow{Y_{2}} \cdots, \tag{8}\]  
where \(\{X_{n},n\geq 0\}\) is a discrete Markov chain where \(X_{n}\) represents the lane state for the \(n\)-th BV, and \(X_{n}\) can be \(l_{1}\) or \(l_{2}\). Transitions \(\{Y_{n},n\geq 0\}\) denote the timer delay between two consecutive BVs; \(Y_{n}\) depends on states \(\{X_{n},X_{n+1}\}\), but it is independent of the previous timer delays \(Y_{k},k<n\).  
Since both \(H_{i}\) and \(T_{\mathrm{L}_{i,j}}\) are random variables that can hardly yield a closed-form expression, an approximation method for the backoff timer delays is based on the expected values of the involved random variables. Based on Equation (4), we have:  
\[E[T^{\mathrm{c}}]=\mathrm{E}[H]\mathrm{E}[T_{\mathrm{L}}], \tag{9}\]  
where \(\mathrm{E}[T^{\mathrm{c}}]\) denotes the expected delay for a generic routing phase out of the \(N_{\mathrm{p}}\) ones that are carried out during the multimedia content delivery, \(\mathrm{E}[H]\) is the average number of hops and \(\mathrm{E}[T_{\mathrm{L}}]\) represents the expected timer delay in one V2V link. \(\mathrm{E}[H]\) can be found as:  
\[\mathrm{E}[H]=d_{\mathrm{SD}}/\mathrm{E}[D_{\mathrm{L}}], \tag{10}\]  
where \(d_{\mathrm{SD}}\) is the end-to-end distance between the SUV and the selected SOV, and \(E[D_{\mathrm{L}}]\) denotes the average transmission distance for each established link between two consecutive BVs.  
The main goal of this section is to derive the probability distributions and mean values of \(T_{\mathrm{L}}\) in Equation (9) and \(D_{\mathrm{L}}\) in Equation (10), respectively.  
The reminder of this section is organized as follows. First, given the current lane state \(X_{n}\) and the next hop lane state \(X_{n+1}\), the conditional distribution of one-hop link delay/distance for the intra-/inter-lane scenario, \(\mathrm{Pr}\{T_{\mathrm{L}}|X_{n},X_{n+1}\}\) and \(\mathrm{Pr}\{D_{\mathrm{L}}|X_{n},X_{n+1}\}\), are derived (Section 5.2). Based on the conditional timer distribution, the distribution of the one-hop timer delay is obtained (Section 5.3). Based on the conditioned distance distribution and the lane state transition matrix, the distribution of one-hop transmission distance is obtained (Section 5.4). Finally, based on these obtained results, we estimate the control delay time in one constructed routing path (Section 5.5).