## 6 Results and Discussions
### Simulation of the Monitoring and Control of Parking Lots in the North of a City
#### 6.2.3 Sensitivity of the M/M/c/N Submodel Parameters  
In this section, the effect of the parameters of the M/M/c/N submodel on the steady-state measures of performance of the parking lot system was studied through a sensitivity analysis. A set of simulation experiments were carried out considering the following parameters: \(\lambda\) = inter-arrival rate (cars/hour); \(\mu\) = 1/average service time (cars/hour); \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{3}{c}{**1000 h**} & \multicolumn{2}{c}{**5000 h**} \\ \hline \multirow{2}{*}{**Measure**} & **Theoretical from** & \multirow{2}{*}{**Experimental**} & \multirow{2}{*}{**Accuracy**} & \multirow{2}{*}{**Experimental**} & **Accuracy (\%)** \\  & **Mathematical Model** & & **(\%)** & & **(\%)** \\ \hline Lq & 6.04420 & 5.91519 & 97.87 & 6.02325 & 99.65 \\ Wq & 0.36579 & 0.35550 & 97.19 & 0.36486 & 98.74 \\ Ls & 55.66485 & 55.45165 & 99.62 & 55.58058 & 99.85 \\ Ws & 3.36879 & 3.32377 & 98.66 & 3.36109 & 99.77 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Estimation of the accuracy of the parking lot model considering the steady-state performance measures. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Arrivals**} \\ \hline Number of arrivals (\#cars) & 20,022 \\ Average arrival intervals (h) & 0.0499459154364 \\ \hline Average arrival rate (\#cars/h) & 20.0216572519182 \\ Number of not waiting customers (\#cars) & 2175 \\ \hline Average nonzero waiting time (h) & 0.408778925789 \\ Number of lost customers (\#cars) & 3326 \\ \hline Probability PN that there are N clients in the system; N = 60, P60 & 0.166117271 \\ \(\lambda_{\text{lost}}\) =Lost customers rate (\(\lambda\) * P60) (\#cars/h) & 3.325943064 \\ \hline Effective arrival rate (\(\lambda_{\text{eff}}=\lambda-\lambda_{\text{lost}}\)) (\#cars/h) & 16.69571419 \\ \hline \multicolumn{2}{c}{**Service**} \\ \hline Average service time (h) & 2.97829065 \\ Average service rate (\#cars/h) & 0.335763065 \\ \hline Number of customers served (\#cars) & 16,636 \\ Average number of busy parking spaces (\#cars) & 49.53645989 \\ \hline Utilization of parking lot (\%) & 99.07291979 \\ \hline \multicolumn{2}{c}{**Queue**} \\ \hline Average queue length (\#cars) & 5.915193287 \\ Average waiting time (h) & 0.355495085 \\ \hline \multicolumn{2}{c}{**System**} \\ \hline Average system size (\#cars) & 55.45165318 \\ Average system time (h) & 3.323770353 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Results of the 1000-h simulation of a parking lot model that includes an M/M/c/N queue system. c = total number of parking spaces (#cars);  
QC = queue capacity (#cars);  
N = system limit (#cars). In addition, measures of performance considered in the analysis were:  
Ls = expected number of customers in system;  
Lq = expected number of customers in queue;  
Ws = expected waiting time in system;  
Wq = expected waiting time in queue. Effect of Changes in the Arrival Rate  
For this analysis we use the following settings:  
\(\lambda\) = [18,19,20,21,22];  
\(\mu\) = 0.333 cars/hour;  
N = [20,40,60,80,100];  
QC = 10. The effect of system parameters on the performance measures is summarized in Table 11. First, we observe that the increase in \(\lambda\) caused an increase in the expected number of customers in queue and in the expected waiting time in queue. Additionally, we observe that the expected number of customers and the expected waiting time in system increased if we increased the value of \(\lambda\). An increase in the number of parking spaces could help to confront the effects of increasing \(\lambda\). However, an uncontrolled increase in the number of parking spaces or the queue could transform them into underused resources.