## 8 Conclusions  
An M/M/1 queuing model based on Kendall notation was proposed to solve the problem that currently exists in the synchronization of traffic lights on 50 th Street in Panama City. The mathematical model included a stability analysis of the system, performing the analysis with two capacities of the system until achieving the stability of the queuing system at 1300 vehicles per hour.  
For the AI component, we measured the accuracy, as shown in Table 9. We are considering evaluating other models to make the AI component even more robust, which will be studied in future research. Evaluating other models can help identify the strengths and weaknesses of the current model and compare its performance with other models in different scenarios. In addition, it can also improve the ability of the current model to make more accurate and valuable predictions in more diverse situations.  
The algorithm developed in MATLAB was based on a stability analysis, which shows the stability and instability for January 2020. The stability analysis found that the system is not saturated on holidays and weekends. On weekdays, we observed that the system is oversaturated with the capacity currently having the traffic light cycle of 50 th Street. With this, we can analyze that the current scheduling system of 50 th Street is an unstable, oversaturated queuing system, which would generate large queues with non-estimated departure times.  
A simulation was carried out in Excel over a single day. The chosen day was at 98% of its capacity. The results obtained are that the approximate duration in the queuing system  
Figure 16: Prediction model for traffic flow intensity demand at 6:00 p.m.  
\begin{table}
\begin{tabular}{c c} \hline
**Traffic Flow Intensity Demand at:** & **Coefficient r\({}^{2}\)** \\ \hline
3:00 p.m. & 0.7148 \\
4:00 p.m. & 0.9713 \\
5:00 p.m. & 0.6423 \\
6:00 p.m. & 0.7262 \\ \hline \end{tabular}
\end{table}
Table 9: Coefficients of determination (r\({}^{2}\)) by traffic flow intensity demand for hours.  
is 0.002 s in the best scenario with 0 elements in the queue and with waiting times of up to 60 min. Regarding the service capacity of the system, several system stability analyses were carried out, where the current traffic light capacity is insufficient for the number of vehicles passing through the road at the 3:00 p.m. peak hour. The first analysis is of a capacity of one thousand vehicles per hour (current capacity), giving us a stability of 22.58% and an instability of 77.42%. With this capacity, the system was so saturated that it was not feasible to apply the model. Starting with a capacity of one thousand vehicles per hour, we continued analyzing one hundred at a time until we reached a capacity of one thousand three hundred vehicles per hour (suggested capacity). With this capacity, we obtained a stability of over 70%. Compared with the previous capacities, this result is feasible, and the system analysis could be carried out using [PERSON]'s notation. For the waiting time in the system, two scenarios were analyzed:  
* Scenario 1. Vehicle capacity of 1000 vehicles per hour; in this scenario, employing only [PERSON]'s notation model, it would be possible to estimate the average waiting values on holidays and weekends in January at 3:00 p.m., where a short queue with an average of 18 vehicles waiting for each cycle and a waiting time of 65 s on average was determined. In the rest of the days, the queuing system becomes unstable, and it would not be possible to estimate the waiting times in the system employing this model.
* Scenario 2. For a capacity of 1300 vehicles per hour, the service time for each vehicle is 37 s, having at least 13 vehicles in the queue in the best-case scenario where the light is green, and no vehicles are stacked in the queue. The capacity was analyzed from 3:00 p.m. to 4:00 p.m. every day of January 2020. It was observed that, at that time, there was more congestion. A vehicle could take between 2 to 60 min to be served.