# Fuzzy-Based Control System for Solar-Powered Bulk Service Queueing Model with Vacation
## 3 Materials and Methods  
### Definitions  
3.1.1. TFN (Triangular Fuzzy Number)  
A type of fuzzy number represented by a triplet (l, m, u), where l is the lower bound, m is the most likely value (mode), and u is the upper bound. It is used to model uncertainty in parameters by assigning a fuzzy range instead of a crisp value. 3.1.2. P-NLP (Parametric Nonlinear Programming)  
This is an optimization method where the objective function or constraints are nonlinear and include parameters (e.g., \(\alpha\)-cuts in fuzzy systems). It is used to compute performance bounds under fuzziness by solving optimization problems at different confidence levels. 3.1.3. FLC (Fuzzy Logic Controller)  
This is a control system based on fuzzy logic that uses linguistic rules rather than mathematical models to map inputs to outputs. It is particularly useful for handling uncertainty and imprecision in complex systems like queuing networks or energy-aware controllers. \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Criteria** & **Fuzzy-Based Control System (FBCS)** & \begin{tabular}{c} **Hybrid Fuzzy Models (e.g., ANFIS, Fuzzy-NN, Sugeno-GA)** \\ \end{tabular} \\ \hline Adaptability & \begin{tabular}{c} High—Dynamically adjusts service rate, bulk size, and vacation based on input data \\ \end{tabular} & \begin{tabular}{c} Moderate—Requires retraining or offline tuning to adapt to changing conditions \\ \end{tabular} \\ \hline Interpretability & \begin{tabular}{c} Excellent—Rule-based Mamdani structure allows transparent decision-making \\ \end{tabular} & \begin{tabular}{c} Low—Operates as a black box; rules and outputs are difficult to trace \\ \end{tabular} \\ \hline Uncertainty Handling & \begin{tabular}{c} Robust—Uses \(\alpha\)-cut analysis with P-NLP to quantify fuzzy output ranges \\ \end{tabular} & \begin{tabular}{c} Limited—Typically uses fixed defuzzification without formal uncertainty bounds \\ \end{tabular} \\ \hline Real-Time Performance & \begin{tabular}{c} Optimized for real-time, adaptive queue control in solar-powered systems \\ \end{tabular} & \begin{tabular}{c} Slower—High computational demand, not always suited for real-time applications \\ \end{tabular} \\ \hline Energy-Aware Control & \begin{tabular}{c} Explicitly integrates solar power availability in decision logic \\ \end{tabular} & \begin{tabular}{c} Generally lacks dedicated energy inputs or control mechanisms \\ \end{tabular} \\ \hline Data Requirements & \begin{tabular}{c} Low—Relies on expert knowledge and simulations rather than large datasets \\ \end{tabular} & \begin{tabular}{c} High—Needs significant training data for model accuracy \\ \end{tabular} \\ \hline Application Suitability & \
\begin{tabular}{c} Ideal for solar-powered queueing and telecom systems under dynamic conditions \\ \end{tabular} &
\begin{tabular}{c} Suitable for static environments with rich data but limited responsiveness \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fuzzy-Based Control System (FBCS) model. To develop a fuzzy-based Bulk Service Queueing Model with Vacation, we consider the following assumptions:  
Customers arrive at the system following a Poisson process with a mean arrival rate \(\chi\). Arrivals are independent and occur randomly over time. The server provides bulk service with a maximum batch size per service. If the queue length is greater than or equal to \(\beta\), the server serves a full batch. If the queue length is between \(\alpha\) and \(\beta\) (\(1\leq\alpha\leq\beta\)), the server initiates service with the available customers. Service time follows a general distribution with mean service rates: High Bulk Service a, Medium Single Service b, and Low Vacation c.  
After completing a service cycle, the server takes a vacation if the queue length is below the threshold b. Vacation time follows a general distribution with mean vacation time \(\theta\). During the vacation period, arriving customers must wait until the server returns.