## 7 Results and Discussions
### Optimization of the Queuing System  
1. Analysis for the month of January for a traffic light capacity of 1000 vehicles/h. The analysis of the first server (traffic light 1) shows the results: stability of 22.58% and instability of 77.42%. The results of the other variables analyzed on the days selected for this server are shown in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**January Day 2020** & **L** & **Lq** & **W** & **Wq** \\ \hline
1 & 1.4510 & 0.8590 & 0.0025 & 0.0015 \\
5 & 7.1301 & 6.2531 & 0.0081 & 0.0071 \\
9 & 5.2112 & 4.3722 & 0.0062 & 0.0052 \\
12 & 20.7391 & 19.7851 & 0.0217 & 0.0207 \\
19 & 13.9254 & 12.9924 & 0.0149 & 0.0139 \\
26 & 9.7527 & 8.8457 & 0.0108 & 0.0098 \\
29 & 499 & 498.002 & 0.5 & 0.499 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of L, Lq W, Wq. For a capacity of one thousand vehicles. First Server. 3:00 p.m.  
Figure 10: Results of the tola50 function. Figure 9: Queuing system at the 50 th Street intersection (Nicanor de Obarrio Ave). In addition, the following information is obtained for this queuing system:  
* The average number of vehicles in the queue is 18 vehicles per light change. * The average time to be served is: 65 s, under the condition that it only occurs on holidays and weekends. * For the capacity of 1000 vehicles per hour, the system was 77% of the time over saturated, with an average occupancy rate of 118%. * The results of the Kendall notation were obtained on holidays and weekends with a value of \(\rho<1\), and an average occupancy of 87% of the system. Using the program, it was found that the average number of cars leaving the system after being served by the server (traffic light 1) corresponds to 20% and 80%. The system has four parallel lanes that continue, and one leaves the main road because the left lane forks. Taking this information into account, the percentage of distribution that will be the restriction to advance to the next server (traffic light 2) is calculated, and 100% of the cars entering the system is divided by five to obtain this percentage. From the results obtained in the first server, 80% of cars waiting to be served by the second server (Traffic Light 2) are considered. In this second server, stability improves, and 64.5% stability and 35.5% instability are obtained. Additionally, the following results are obtained:  
* The average number of vehicles in the queue is 11 vehicles per light change. * The average time to be served is 40 s, under the condition of stability on holidays and weekends. * For the capacity of 1000 vehicles per hour, the system was 65% stable and 35% unstable. * For the full month, the system had an occupancy level of 94%, which confirms the improvement in stability. * The system was at an average occupancy of 85% on days where the value of \(\rho\) is less than 1, i.e., \(\rho<1\). Additionally, stability and instability values were obtained for the capacity of 1000 vehicles at the 4:00 p.m., 5:00 p.m. and 6:00 p.m. schedules. The results obtained are shown in Table 2:  
From this second traffic light, we have two outputs. The second output corresponds to cars leaving the second server and leaving the system (see Figure 11), and the third output is not considered in this study. 2. Analysis for the month of January for a traffic light capacity of 1300 vehicles/h. After several tests for the traffic light capacity of vehicles per hour, it was determined that the value of 1300 vehicles per hour for \(\mu\) is where a stability of more than 70% is achieved. An improvement in stability of 77.4% was obtained and in instability of 22.6%. By having a stable queuing system, it was possible to calculate the average values by means of Kendall notation with a more accurate study of the system. For the first server (traffic light 1) the following results were obtained:* The average number of vehicles in the queue is 13 vehicles per light change. * The average time to be served is 37 s, if and only if the vehicle arrives when the light is green and there is no vehicle waiting.