## 5 Numerical and Simulation Results
### Simulation Environment  
The simulation environment is VANET with MATLAB. Since the focus of these relay-node selection methods lies on link level, just 802.11p MAC layer is simulated in this paper. The major communication parameters are identical to those used in [13; 14; 19], listed in Table 2. Since most distance between adjacent intersections are in the range of \((200,800)\) m, \(R\) is set as 400 m, a small value for \(R\) in VANET.  
Each branch has a length of 700 m to ensure the simulation zone long enough for two hops. To assure nodes available in the partition range and evaluate the enhancement of our proposal in heavy traffic, the vehicle density \(\lambda\) for the simulation to compare the performance in Section 5.3 is set from 0.0025 to 3.0025 vehicle/meter at the interval of 0.2 vehicle/meter. Vehicles are distributed on every branch randomly, following Poisson distribution of the density \(\lambda\). For simple simulation without loss of generality, the Hunter is located at the position \(R/2\) far from C on the eastern branch in each simulation.  
In the simulation, the value of the maximum speed \(v_{\text{max}}\) of vehicles is determined with the value of the distance between the adjacent vehicles to comply with the rule related to the safe inter-vehicle distance [26; 27]. \(v_{\text{max}}\) can be given as  
\[v_{\text{max}}=min(d_{\text{inter\_veh}},v_{\text{max\_rule}}) \tag{31}\]  
where \(d_{\text{inter\_veh}}\) is the average inter-vehicle distance, and \(v_{\text{max\_rule}}\) represents the limit speed in a specific road scenario. The units of \(v_{\text{max}}\) and \(d_{\text{inter\_veh}}\) are km/h and m, respectively. Each vehicle chooses a random speed following a uniform distribution in \((\frac{1}{2}v_{\text{max}},v_{\text{max}})\) at the beginning of the simulation and keeps the chose speed during the simulation. Lane change and overtaking are not modeled for vehicle movement. From the simulation results, the duration for the message going through the intersection range is less than 3 ms. Moreover, the minor impact of node mobility on the relay-node selection has been proved in Ref. [28; 29]. Hence, the above assumptions about the vehicle movement and the adoption of MATLAB are acceptable.  
The arrival rate of messages is set to 2 EMs/s, \(N_{\text{recon}}=3\) for the validation of the PDR model to separate the curves and \(N_{\text{recon}}\) for others. \(C_{W(1)}\) is chose as 1 for AMB because of a few candidates in the CTB contention, and 4 for others. \(T_{\text{m\_slot}}\) in mini-BBM and \(T_{\text{inter}}\) in CTB contention are selected as \(T_{\text{inter}}/3\) and \(T_{\text{inter}}/2\) to reduce the partition latency in sparse vehicles and the contention latency in heavy traffic, meanwhile to avoid the spurious forwarding [30].  
We performed 20,000 repetitions of Monte Carlo simulation [31; 32] for PDR results and 1000 repetitions for other results to get statistical significance. These outcomes are averaged to produce the graphs presented in this section with 95% confidence intervals. The confidence intervals are marked with the error bars in the plots.