# Fuzzy-Based Control System for Solar-Powered Bulk Service Queueing Model with Vacation
## 4 Results and Discussion  
### Numerical Results and Cost Optimization with Solar Power System Example  
In this section, the accuracy and applicability of the analytical results derived in the previous section are verified through numerical examples. A comprehensive numerical experiment is performed to investigate the effects of key parameter values on the proposed model. The total cost function (TC) is computed using three distinct cost sets, as shown in Table 3. The numerical outcomes for various performance metrics and cost analysis are summarized in Tables 4 and 5.  
Figure 3: Crisp performance output.  
Consider a solar-powered telecommunications base station (BTS) that operates under a bulk service queueing system. The arrival rate of energy demands (\(\lambda\)) and service rates (a, b, c) are modeled as trapezoidal fuzzy numbers (TFNs) to account for variability in sunlight availability and energy storage constraints. The cost elements in Table 1 now represent the operational and maintenance costs of different power setups:  
**Cost Set I:** Traditional grid-powered BTS with diesel backup  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(\mathbf{\alpha}\) & \(\mathbf{x_{1,\alpha}^{LB}}\) & \(\mathbf{x_{1,\alpha}^{UB}}\) & \(\mathbf{x_{2,\alpha}^{LB}}\) & \(\mathbf{x_{2,\alpha}^{UB}}\) & \(\mathbf{x_{3,\alpha}^{LB}}\) & \(\mathbf{x_{3,\alpha}^{UB}}\) & \(\mathbf{x_{4,\alpha}^{LB}}\) & \(\mathbf{x_{4,\alpha}^{LIB}}\) \\ \hline
0 & 2.000 & 3.50 & 6.000 & 7.000 & 10.000 & 11.00 & 7.000 & 8.000 \\ \hline
0.1 & 2.025 & 3.40 & 6.950 & 6.950 & 10.025 & 10.98 & 7.025 & 7.950 \\ \hline
0.2 & 2.050 & 3.30 & 6.900 & 6.900 & 10.050 & 10.97 & 7.050 & 7.900 \\ \hline
0.3 & 2.075 & 3.20 & 6.850 & 6.850 & 10.075 & 10.85 & 7.075 & 7.850 \\ \hline
0.4 & 3.000 & 3.10 & 6.800 & 6.800 & 10.100 & 10.80 & 7.100 & 7.800 \\ \hline
0.5 & 3.025 & 3.00 & 6.750 & 6.750 & 10.125 & 10.75 & 7.125 & 7.750 \\ \hline
0.6 & 3.050 & 3.90 & 6.700 & 6.700 & 10.150 & 10.70 & 7.150 & 7.700 \\ \hline
0.7 & 3.075 & 3.80 & 6.650 & 6.650 & 10.175 & 10.65 & 7.175 & 7.650 \\ \hline
0.8 & 3.100 & 3.70 & 6.600 & 6.600 & 10.200 & 10.60 & 7.200 & 7.600 \\ \hline
0.9 & 3.125 & 4.60 & 6.505 & 6.550 & 10.225 & 10.55 & 7.225 & 7.550 \\ \hline
1 & 3.175 & 4.50 & 6.500 & 6.500 & 10.250 & 10.50 & 7.250 & 7.500 \\ \hline \hline \end{tabular}
\end{table}
Table 4: \(\alpha\)-cuts of fuzzified arrival rate (\(\lambda\)) and service rates (a, b, c).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Cost Set**} & **Infrastructure** & **Battery** & **Grid** & **Maintenance** & **Solar Panel** & **Diesel Fuel** & **Storage** & **System Monitoring** \\  & **Cost** & **Cost** & **Power Cost** & **Cost** & **Cost** & **Cost** & **Cost** & **Cost** \\ \hline
**I** & 140 & 80 & 60 & 30 & 25 & 15 & 30 & 20 \\ \hline
**II** & 120 & 90 & 50 & 20 & 15 & 20 & 10 & 5 \\ \hline
**III** & 100 & 60 & 30 & 10 & 10 & 10 & 6 & 4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Cost sets for energy management in a solar-powered BTS.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{\alpha}\) & \(\mathbf{(EL_{s})_{a}^{LB}}\) & \(\mathbf{(EL_{s})_{a}^{UB}}\) & \(\mathbf{(EW_{s})_{a}^{LB}}\) & \(\mathbf{(EW_{s})_{a}^{UB}}\) & \(\mathbf{(TH)_{a}^{LB}}\) & \(\mathbf{(TH)_{a}^{UB}}\) \\ \hline
0 & 1.370 & 16.014 & 0.408 & 3.203 & 0.960  
**Cost Set II:** Hybrid solar-diesel system  
**Cost Set III:** Fully solar-powered system with battery storage