## 3 Results and Discussions
### Effects on Traffic Flow  
The flow variation is investigated for several values of \(\alpha\), as a function of the vehicles' extraction \(\beta\) and the pedestrian entrance \(\alpha_{p}\), to understand the effect of the crosswalk. As observed in Figure 3, the flow decreases quantitatively as \(\alpha_{p}\) increases. However, the plateau region decreases qualitatively as \(\alpha\) increases. These effects suggest the crosswalk induces a bottleneck. The system's phase diagram (the construction of the phase diagram is explained in Appendix A) is constructed on the plane (\(\alpha_{p},\beta\)), taking the vehicles' flow as an order parameter, to achieve a better understanding of the bottleneck effect on traffic flow. In Figure 4, it can be observed that the system reaches up to four phases: maximum current (MC), jamming (J), congestion (C), and gridlock (GL). These four phases depict all states in which traffic flow is present in the system. In this way, maximum current describes steady to limited-steady traffic, jamming phase describes high-density steady traffic to traffic at saturation, congestion describes the state with unstable speed, and gridlock describes the state when traffic is completely standstill. For each phase, more detailed descriptions are given below. The maximum current phase is characterized by presenting free-flowing conditions, which means vehicles can move at their desired speed, except for the small platoons of standing vehicles that appear in the vicinity prior to the crosswalk and at the end of the lane (see Figure 5a). In addition, in this phase, it is common to observe gaps between vehicles, and these gaps are used by pedestrians to traverse the crosswalk without perturbing traffic flow. As depicted in Figure 4a, for \(\alpha=0.1\), the maximum current is the largest phase where the flow in the lane becomes decoupled from the pedestrian entrance and extraction rates, i.e., in this phase, the flow does not depend on \(\alpha_{p}\) and \(\beta\). When \(\alpha=0.4\), the maximum current phase is reached when \(\beta>0.5\) and \(\alpha_{p}<0.3\) and is observed as a planar surface in Figure 3b. The range of values for \(\alpha_{p}\) and \(\beta\) of the maximum current phase decreasesaccording to the increase in \(\alpha\); hence, for \(\alpha=0.7\) this phase is bounded by \(\alpha_{p}<0.1\) and \(\beta>0.7\) (see Figure 4c). As the queue prior to the crosswalk increases, the system transitions to the jamming phase or to the congestion phase, depending on the vehicles' extraction rates \(\beta\). The jamming phase appears for pedestrian entrance rates \(0\leq\alpha_{p}<0.72\), increasing its extension, with respect to \(\beta\), as the injection rate \(\alpha\) increases. Indeed, the extension of the jamming phase increases according to the decrease in the maximum current phase as depicted in Figure 4b-d. Here, the effects of the extraction and the bottleneck are combined, inducing a queue of stop-and-go vehicles all along the lane as depicted in Figure 5a. Due to this condition, the queue is observed to be dense enough with closely spaced vehicles along the lane. In this phase, vehicles still interact among them, while the pedestrians' influx does not completely inhibit the traffic flow. If the pedestrian entrance \(\alpha_{p}\) increases, triggering the rapid growth of both the bottleneck and the queue prior to the crosswalk, then the system transitions to the congestion phase. Congestion takes place when the bottleneck worsen the system's traffic flow. As depicted in Figure 5c, the crosswalk becomes a considerable obstruction strongly influenced by pedestrians occupying the crosswalk, generating long queues of non-moving vehicles. In this phase, the traffic flow decreases as \(\alpha_{p}\) increases (see Figure 3), which means the congestion phase is expanded as the maximum current phase is contracted (see Figure 4).