# The Effect of Speed Humps on Instantaneous Traffic Emissions
## 4 Results and Discussion
### The Impact of Speed Humps on Average Velocity and Traffic Flow  
In this section, we present results of the effect of speed humps on average velocity and traffic flow. In Figure 3 we show the average velocity as a function of density for different number of speed humps, notice that all the curves follow the same trend. For low densities \(0\leq\rho\leq 0.12\), observe that the average velocity is just below \(V_{\max}=4\) for all the cases but remains almost constant, this is due to the effect of the braking probability \(P=0.1\) which reduces the average velocity. Then for \(\rho\geq 0.12\) the average velocity decreases rapidly and the curves overlap as density increase, this suggests that for high densities of vehicles the number of speed humps has no significant impact on velocity. This effect is due to the large number of interactions between vehicles, they need to decelerate to avoid collisions and accelerate as soon as there is space to move forward. The average velocity goes to zero due to a rising number of local congestions as we increase the number of vehicles on the lane. Figure 2: A sketch of the model, where the road is represented by an array of \(L=1500\) equivalent to a lane of 11.25 km. Here, \(d_{n}\) is the spatial-headway and \(D_{d}\) is the number of empty cells between a vehicle and the closest speed hump ahead. In Figure 4 we present the fundamental diagram for different number of speed humps. We observe that all the studied cases follow the same trend with three observable phases. The free-flow phase for \(0\leq\rho\leq 0.12\) where we observe an increase on flow for all the cases. Please note that the cases with more speed humps have a lower slope that those with low number of speed humps. Note the slight difference of the curves with larger number of speed humps from the one with only one speed hump, this suggests a low influence of speed humps on the flow in this phase. The second phase for \(0.12<\rho\leq 0.48\) shows a steady phase with no significant variations on the flow for the simulated cases. The third phase for \(0.48<\rho\leq 1\) shows a gradual decrease on the flow for all the studied cases. Observe the difference of the curves from 0.48 to 0.7 due to the influence of speed humps, the curves are smoother for the cases with larger number of speed humps. However, for \(\rho\geq 0.7\) the curves overlap, this suggests that the influence of speed humps on flow has vanished, therefore, the decrease on flow results from local congestions due to the number of vehicles on the lane. To study the effect of a large number of speed humps on flow and velocity we performed simulations with more than 30 speed humps. In Figure 5 we present the flow and velocity as a function of the number of speed humps. We took three density values to represent each phase: \(\rho=0.1\) for the first phase, \(\rho=0.3\) for the second phase, and \(\rho=0.8\) for the third phase. We tested with up to 100 equidistant speed humps. Observe that for \(\rho=0.1\) there is a downward trend for flow and velocity, both presenting fluctuations as the number of speed humps increases. For the case \(\rho=0.3\) the flow presents a very low decrease with the rise on the number of speed humps, the velocity shows a constant value with the increase on the number of speed humps. Finally, for \(\rho=0.8\) we observe a steady behavior; no changes for the flow nor the velocity. We use Equation (6) to compute the standard deviation of velocity, where \(v_{i}\) is the velocity of the \(i\)-th vehicle on the lane, \(\bar{v}\) is the average velocity and \(N\) is the number of vehicles on the lane. \[\sigma=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(v_{i}-\bar{v})^{2}} \tag{6}\]  
Figure 3: Average velocity as a function of density. The velocity indicates the average number of cells to move forward per simulation step, and the density \(\rho=N/L\). In Table 1 we present the results for \(\rho=0.1\) for a selected number of speed humps.