14 Apr 3

Queuing Theory

This is the theoretical foundation of the study. The thesis analyzes the queuing phenomenon in school canteens by introducing the $G/M/c$ model, and tries to quantify key indicators such as students’ arrival rate, average waiting time, and the number of service windows from the perspective of mathematical modeling, to optimize the queuing system.

$G/M/c$ Model

It refers to arrival time obeying General distribution, the service time is a Markovian (exponential) distribution, and the queuing model with c service windows. The model is used to simulate complex queuing situations in reality, such as a cafeteria with multiple window queues, and can reflect the congestion of the service system during peak hours. Related to it are a series of sub-metrics (e.g., $\lambda$, $\mu$, $W_q$, $L_q$, $\rho$, etc.), which are the core parameters for calculating the queuing performance.

Crowding Dynamics

Refers to the overall dynamic process of students entering, staying, and queuing in the cafeteria. This term covers the examination of peak hour flow analysis, different student behavioral patterns (e.g., just passing by vs. actual diners), indoor and outdoor dining choices, etc., and is what this study hopes to explain and optimize.

Theory

Coefficient of Variation ($C_a$)

The coefficient of arrival time variation is used to assess the volatility of student arrival to the cafeteria. Because student arrival times do not obey the full Poisson distribution, $C_a$ is used to correct the $M/M/c$ model to be closer to the actual situation, thus transitioning to the $G/M/c$ model. It is an important statistical correction factor for performing advanced modeling.

Arrival Rate ($\lambda$) vs. Service Rate ($\mu$)

Indicates the rate at which students enter the system and are served per unit of time. These two parameters directly determine the utilization rate $\rho$ and queuing status of the system, which is the first step in constructing the model and the core data to regulate the efficiency of the system. The thesis provides a careful estimation of these two parameters through field observations and time-stamped records.