Introduction
In my own experience of our school, I often had to spend over fifteen minutes in line,
only to wait for food. This personal frustration led me to question whether these delays were
simply due to high demand or rooted in deeper systemic inefficiencies. Long queue times and
overcrowding in school cafeterias are known to be serious problems facing students. This issue
has been well documented in previous studies that have emphasized the impact of poorly
designed cafeterias on student satisfaction, waiting times, and the overall dining experience
(Galabo, 2019). Surveys conducted at our school have shown that many students are
dissatisfied with the current cafeteria setup, citing overcrowding and long waiting times as
major annoyances. First, we break down the students’ dissatisfaction into two main areas.
Firstly, the long waiting time consumes their patience, which is totally unnecessary. Second,
the crowded cafeteria makes it difficult for them to move around.
However, no one has fully explored why these issues persist after years of improvement
or how specific factors, such as cafeteria layout, queueing systems, and crowding during peak
hours, contribute to the problem. If we can identify the root causes of inefficient cafeteria
operations, we can not only improve the dining experience but also optimize space usage and
human resources. This study aims to investigate the root causes of cafeteria crowding and long
waiting times by analyzing queue dynamics, conducting simulations, and exploring potential
improvements. By identifying and addressing these issues, I hope to reduce waiting times and
crowding situations to increase student satisfaction and to improve overall operational
efficiency. And I also hope to use this analysis to provide a generic template and quantitative
solution for other similar situations.
Literature Review
To understand the mechanisms behind these inefficiencies, it is essential to examine
existing theoretical frameworks and studies related to queueing behavior. One of the most
widely used approaches in this area is queueing theory, which provides mathematical tools to
analyze waiting lines. Queueing theory, a subfield of operations research, focuses on analyzing
and optimizing systems where entities wait in line for service. The foundational models, such
as M /M /1, M /M /c, and M /D/1, have been widely adopted to understand arrival patterns,2
service rates, and queue dynamics under probabilistic assumptions (Kambli et al., 2020). These
models are defined by Kendall’s notation, where ”M” refers to Markovian (Poisson) arrival
and/or service times, and the number indicates the quantity of parallel servers (Ajiboye &
Saminu, 2018). Little’s Law, a fundamental result in queueing theory, relates average waiting
time to arrival rate and system capacity (Tang et al., 2019).
In the context of campus cafeterias, queueing theory has been extensively employed to
address issues of student congestion during lunch hours, which are particularly pronounced due
to synchronized class schedules (Zuo, 2020). The application of models such as M/M/n/m has
shown practical potential in optimizing the number of open windows and reducing average
waiting time by reallocating resources during peak demand (Chen & Wang, 2011). Simulation
studies demonstrate that queue optimization can reduce waiting times by nearly 45% (Ye,
2009). Another study using a multi-stage queue model reduced average customer dwell time
from over 500 minutes to under 15 minutes by redesigning queue architecture and service logic
(Ajiboye & Saminu, 2018).
This methodology was selected because it enables the formulation of a structured,
quantitative framework to address crowding issues through measurable parameters such as
arrival rate, service rate, and waiting time. Unlike anecdotal methods, queueing theory allows
for reproducible simulation and modeling, which can be validated and iteratively refined.
Additionally, this approach provides rapid validation of the solution. In most cases, it is
sufficient to modify the input variables and compare the output with the previous one.
Research Gap
Despite these successes, classical queueing models exhibit limitations when applied to
real-world cafeteria environments. Most models assume steady-state behavior, homogeneous
service stations, and exponentially distributed arrival and service times, which rarely match the
fluctuating and clustered nature of student behavior (Lu et al., 2021). Furthermore, traditional
models fail to account for human behavioral dynamics, such as balking, reneging, or jockeying,
where students leave or switch queues due to perceived delays (Chen and Wang, 2011).
Extensions to these models have attempted to incorporate dynamic arrival rates and customer
patience thresholds (Li & Saminu, 2016). Moreover, complex queue networks involving3
multiple food stations and checkout counters require simulation or agent-based modeling, as
closed-form analytical solutions are infeasible (Kambli et al., 2020; Deng et al., 2011).
And there is no suitable theoretical model to describe dynamic variables, such as, in this
study, arrival rates that change over time (students arrive in a concentrated manner during peak
hours, followed by a gradual decrease in arrival rates). In this study, an attempt is made to input
dynamic variables over time into the model by replacing static variables with dynamic
variables and eventually obtaining outputs that vary over time. In other words, all the variables
of the original model are upgraded to time-varying functions, so the final results obtained are
also in functional form with the independent variable time. Also, in this study, there is a
simplified analysis of student behavior. This can be highly integrated with the queueing theory
model and applies to a very wide range of scenarios.
Methodology
Spatial Layout of the Cafeteria
Figure 1 illustrates the layout of the cafeteria used in this study. The topmost area is the
dining area and the remaining area is the working area. There are two entrances to the dining
area, which are located on the upper left and lower left. It is important to note that there are two
dining areas in total. The indoor dining area is shown in the figure. There is also an outdoor
dining area (not shown), which can be reached by leaving through the lower left entrance.
Figure 1: Layout of school cafeteria4
Data Collection
The data for this study came from field observations in our school’s cafeteria, and data
were collected from December 2 to December 5, 2024, and December 9 to December 12, 2024
(8 days in total). Data collection under each schedule lasted for four days due to the school’s
schedule of one cycle every two days. The data primarily covered the third and fourth periods
which are lunch periods, and the peak hours. Specifically, it includes the number of students
entering/leaving the cafeteria during each period, the time taken for services, and the number of
windows.
Data collection was performed through the cafeteria’s monitoring system during the
daytime hours, which recorded the timestamp of each student’s entry into the cafeteria. The
specific time period for data collected was from 11:50 to 13:35 each day. During this time
period, the number of students entering the cafeteria was recorded by manual count, and the
service time of each service was recorded by simple observations.
Ethical Considerations
Data collection was conducted solely for research purposes with the approval of the
school administration. No personally identifiable information of students was collected; only
the data such as entry and exit counts and service times were recorded. Observations were
carried out in a non-intrusive manner to avoid interfering with students’ normal activities. So
the study complies with institutional ethical guidelines for research involving human subjects.
Preprocessing and Entry-Exit Analysis
During the data preprocessing process, all the collected data met the preset criteria, so
no data was eliminated. During the data preprocessing stage, timestamps were converted to
relative times with respect to peak hours and grouped for counting at 2-minute intervals during
peak hours. For off-peak hours, group counts were performed at 5-minute intervals.
We get the average Entries and Exits (Et and Lt) by simply adding them together and
divided by the number of days. Notice that our school’s cafeteria has 2 entrances (marked as A5
and B here). So for each time interval t,
EnterTotal =
8
i=1 Enter ADayi
8 +
8
i=1 Enter BDayi
It is the same way to calculate LeaveTotal.
Figure 2: Average Enter vs Leave Counts Figure 3: Average Net Population
According to Figure 2, the results of the data analysis showed that the arrival time of
students to the cafeteria showed obvious peak characteristics, especially between 11:50 and
12:10 (0 min - 20 min), when an average of about 20 people entered the cafeteria per minute,
while the number of arrivals decreased significantly at other times.
We can also graph the net population in the cafeteria. According to Figure 3, similarly,
there are particularly large numbers of students at the starts of the 2 launch periods, and then
the numbers drop off over time.
Arrival Rate Distribution
Once the data collection was complete we next used the data to calculate the student
arrival rate, which is a variable that changes over time during every lunch time. Then we
consider the following factors. We know that not all of the students who enter the cafeteria will
have any food, and that some may only stay for a short time before leaving (e.g., using the
cafeteria as a hallway just to walk through it). In addition to this, our school has an outdoor
dining area. It has about the same capacity as an indoor cafeteria. Some people order their
lunch and go straight to the outdoor dining area, while others stay indoors to eat their lunch.
We will mainly focus on those students who stay indoors.
According to Table 1, we have a fraction p of the entrants seek service, while the others6
Symbol Description Unit
Et Total number of individuals entering the location dur-
ing interval t
Count
Lt Total number of individuals leaving the location dur-
ing interval t
Count
p(t) Probability that an entering individual seeks service Fraction
r Probability that a service seeker stays after service Fraction
St Number of individuals arriving for service during in-
terval t
Count
λ(t) Average arrival rate of customers during interval t People/s
Table 1: Definitions of variables used in the behavior analysis
leave immediately. After services, some service seekers leave immediately as well, and some
stay after the service. The fraction r of service seekers stay in the area. However, r does not
vary roughly with the time a day because people will always look for uncrowded locations. In
this case, we suggest that they prefer to stay indoors when it is crowded outdoors, and they
prefer to go outside when it is crowded indoors. But it has been observed that it increases
significantly during the cold season, when people are reluctant to have lunch outdoors where it
is colder. So in winter, it usually gets more crowded indoors.
Individual Behavioral Performance Breakdown
Figure 4: Student Behavioral Flowchart7
In this case, we assume the simplest ”ideal man” behavior. According to Figure 4, we
assume that there are only two cases:
- After entering, individuals who do not purchase food (proportion 1− p(t)) leave
immediately, with their presence time considered negligible;
- After entering, individuals who proceed to purchase food(proportion p(t)) remain in the
cafeteria for a period before departing. This period consists of service time T and dining
time in the cafeteria D. Assuming a proportion r of individuals choose to eat indoors, the
total residence time can be approximated as
τnet ≈ T + rD. (1)
Since we only have external data (entry rate Et and exit rate Lt in a given time period t), we
cannot get the number of people who want to be served at a given time period because some of
them are just passing through. So we set up several scenarios: the portion of the population that
buys meals and eats them in the cafeteria leaves after about τ1 = T + D seconds; the portion of
the population that buys a meal and leaves immediately after about τ2 = T seconds; and the
portion of the population that does not buy a meal enters and exits almost instantaneously.
From this, a conservation relation can be formulated:
Lt ≈ 1− p(t) × Et
- r × p(t− τ1) × Et−τ1
Immediately leave after entering
Have finished the meal and leave
- (1− r) × St−τ2
Have purchased food and now leave
. (2)
In the real situation, however, p(t) does not change rapidly. In previous observations, the
proportion of people who went through the cafeteria only decreased slightly over time. So, it is
possible to assume that p(t− τ1) ≈ p(t) since τ1 is not that long. This gives:
L(t) ≈ 1− p(t) Et + p(t) Et−τ1 + r St−τ2.8
Simplify the equation and replace St−τ2 with p(t)Et−τ2 ,
Lt = Et− p(t)Et + p(t)Et−τ1 + r p(t)Et−τ2
= Et + p(t) Et−τ1− Et + r Et−τ2.
Thus, an approximate solution can be obtained:
Lt− Et
p(t) ≈
Et−τ1− Et + rEt−τ2
. (3)
Again, based on the approximation in Equation (1), we combine τ1 and τ2 into a single term
τnet,
Lt− Et
p(t) ≈
. (4)
Et−τnet− Et
According to p(t) = St
Et , we get
St
Lt− Et
≈
.
Et
Et−τnet− Et
The average time high school students spend actually eating lunch is between 7 and 10 minutes
(Lee & Shanklin, 2002). So we can calculate
τnet ≈ 8.5 × 60 × r + T = 510 r + T.
For r we use the indoor and outdoor seating capacity to estimates directly, since people usually
go to places where seats are more available.
Nindoor
47
r =
≈
≈ 0.443.
Noutdoor + Nindoor
59 + 47
Finally we divide S by t to get the student arrival rate:
¯
λt =
St
tperiod
.9
Figure 5: Arrival rate (λ) & Standard Deviation (σ)
The resulting data is shown in Figure 5. We first calculate the average customer arrival
rate¯
λ using the data E and L averaged over multiple days. After that we calculate λ for each
day’s data separately and differ it from¯
λ to get the standard deviation σ from this formula:
σt =
For this type of data we approximate it as a normal distribution, i.e., in this form
1
n− 1
n
i=1
λt,i−
¯
λt
2
Xt ∼ N (λ(t), σ(t)), ∀t ∈ [0, 6300].
Smoothing Spline Fitting
By converting discrete estimates into continuous representations, this method enables
more accurate simulation and forecasting optimization within the Queueing theory framework
(Pollock, 1993). To construct a continuous function for the average arrival rate λ(t) as well as
its standard deviation σ(t), this study employs the Smoothing Spline Fitting (SSF) method. The
method is based on a set of discrete time points {ti}n
i=1 with corresponding values {yi}n
i=1, and
fits a smooth and continuous function f (t) by controlling the degree of bending of the function
while minimizing the residuals of the data (Pollock, 1993). This is expressed mathematically as10
the following optimization problem:
min
f ∈C2
n
i=1
(yi− f (ti))2 + η
b
a
(f′′(t))2 dt .
Here, C2 denotes the function space of twice continuously differentiable functions; yi
represents the target value computed for the i-th time interval; f (t) is the continuous function
to be fitted; and η is the smoothing factor, which controls the trade-off between fitting accuracy
and smoothness (Wahba, 1990). When η = 0, the fitted function passes exactly through all data
points (i.e., interpolation); as λ → ∞, the fitted function approaches linearity (Reinsch, 1967).
In this study, we employed the UnivariateSpline function from the
scipy.interpolate module in Python to fit the functions λ(t) and σ(t) (Virtanen et al.,
2020). This function constructs a cubic spline based on the input data, applies the default
smoothing factor, and satisfies the following optimization formulation:
µ(t) ≈
n
i=0
ciBi(t).
The resulting function f (t) can be evaluated at any timestamp t, thereby generating continuous
representations of the two functions λ(t) and σ(t). These continuous functions are used to
characterize the dynamic customer arrival distribution within the queueing theory model.
Service Rate Distribution
For the service rate data, this study recorded the durations of 100 service sessions
during the lunch period from the surveillance. The data are shown in Figure 6 and the statistics
are shown in Table 2
Service rate refers to the number of services completed per unit of time (number per
second). What is recorded here is the time taken for each service. So we need to take the
reciprocal of the service duration recorded here. The average service rate is¯
µ ≈ 0.030885 (i.e.,
on average, people spend 32.378 seconds during the services). The K-S
(Kolmogorov-Smirnov) test was used in this study to test whether this data can be
approximated to a certain distribution (Lilliefors, 1969). Based on the graph, we determine that11
Figure 6: Service Time Data (1/µ)
it fits the right-skewed characteristic of the exponential distribution very well.
Table 2: Statistics of the Service Time Data
Variable Value
Sample size 100
mean 32.378
std 20.478926
λ 0.030885
D 0.3510
Pval 0.1045
Therefore, we first used the scipy.stats.kstest function in Python to test whether
the sample conforms to an exponential distribution (Virtanen et al., 2020). According to
Table 2, the K-S test yielded a p-value of 0.1045, which is greater than the significance level
α = 0.05. This indicates that we fail to reject the null hypothesis that the service time follows
an exponential distribution, thereby supporting the rationality of approximating the data as an
exponential distribution.
Model Establishment
Based on the previous examination of the data, we believe that the arrival rate conforms
to a normal distribution and the service rate conforms to an exponential distribution (i.e., the12
service process is memoryless). At the same time, the arrival rate varies over time, so the
G(t)/M /c queueing theory model is chosen to describe it, where G(t) denotes a general
time-dependent arrival process, M indicates a service time distribution that satisfies a Markov
process (exponential distribution), and c represents the number of parallel servers (service
windows) (Adan & Resing, 2002).
Basic parameters
Table 3: Variables in the model
Symbol Explanation Unit
λ(t) Average arrival rate of customers people/s
µ Average service rate of a single service window services/s
c Number of service windows Number
ρ(t) System utilization rate /
Ca(t) Coefficient of variation /
Wq (t) Mean waiting time s
Lq (t) Average number of people in the queue Number
P0(t) Probability that the system is empty /
All the variables are shown in Table 3, which the System utilization rate (Kleinrock,
1975):
ρ(t) = λ(t)
cµ
, ρ < 1 (Condition of system stability). (5)
The coefficient of Variation of the arrival time distribution Ca(t) can be calculated bu the
equation:
Ca(t) = σa(t)
1/λ(t), (6)
where σa(t) denotes the standard deviation of the arrival rate computed above. We use Ca(t),
the coefficient of variation of the arrival time distribution, to adjust the M /M /c formulas and
thus obtain results for the G/M /c model (Gross & Harris, 1998).13
Model Computations
Using the generalized form of the Pollaczek–Khinchin (P–K) formula, the mean
waiting time in queue is given by (Whitt, 1993):
Wq =
C2
a (t) + 1
2· Wq, M /M /c, (7)
where Wq, M /M /c is the mean waiting time under the M /M /c model (Little, 1961):
Lq, M /M /c
Wq, M /M /c =
λ . (8)
The mean queue length Lq can also be calculated:
Lq =
C2
a (t) + 1
2· Lq, M /M /c. (9)
Here, Lq, M /M /c (the mean number in queue) is given by the Erlang–C formula (Cooper, 1981):
P0
(λ/µ)c
c!
cρ
(1−ρ)2
Lq, M /M /c =
, (10)
c−1
k=0
(λ/µ)k
k! + (λ/µ)c
c!
1
1− ρ
where the idle-system probability P0 is
P0 =
c−1
k=0
(λ/µ)k
k! + (λ/µ)c
c!
−1
1
1− ρ
Finally, the mean system response time W and the mean number in system L are
. (11)
1
λ
W= Wq +
, L= Lq +
. (12)
µ
µ
Results and Findings
These output variables are then plotted for visualization. According to Figure 7, it can
be observed that during 0 - 290 seconds, the system stability level ρ rises significantly,
exceeding 1, the service capacity is not stable. During this peak period (about the first five14
Figure 7: System Stability During Peak Hour Figure 8: Model Result (W and L)
minutes), the student arrival rate exceeds the service rate, resulting in a situation where the
service can never be completed. Under such circumstances, the system becomes unstable, and
both the number of people waiting and the waiting time grow infinitely, with no steady-state
behavior (see Figure 8).
From Equation (5), we know that the condition for system stability is:
λ
< 1, or equivalently, λ < cµ.
cµ
Therefore, based on the collected data in this study and assuming the number of service
windows and the service rate remain unchanged, the arrival rate must satisfy the following
condition in order for the system to remain stable:
λ < 3 × 0.030885 ⇒ λ < 0.09.
Potential optimization
For the optimization, due to spatial constraints within the cafeteria layout, a maximum
of only four service windows can be opened. Moreover, communications between servers and
students during the service process can significantly prolong service time, thereby reducing the
service rate. Assuming an ideal scenario in which the average service time is reduced to 25
seconds (Reduced communications), the arrival rate must still satisfy:
λ < 4 × 0.04 ⇒ λ < 0.16.15
Figure 9: Optimized Result (W and L)
Based on these considerations, a preliminary optimization strategy is proposed. We
need to limit the arrival rate anyway because there are still some times that the arrival rate is
greater than 0.16 (see Figure 5), and the ideal limit is to control the arrival rate below 0.12
(make sure that the system stability rate ρ is not too close to 1). Given that the proportion of
students choosing to remain in the cafeteria (r) tends to increase during winter (Students don’t
prefer to eat outside because it gets cold), more students would be accommodated within the
cafeteria. This data was also collected during the winter months. Generally, the number of
people in the cafeteria decreases significantly during the summer months. As such, seasonally
adapted control should be implemented: during winter, the arrival rate should be limited to
0.12, and during summer, the limit can be reduced, but it should still not exceed 0.14. As
illustrated in Figure 9, if the arrival rate is restricted to below 0.12, the maximum waiting time
stabilizes around 50 seconds, which falls within an acceptable and efficient operational range.
For a solution to control arrival rates, we need to develop a method that is the most
effective. This would include having a teacher or student volunteer stand somewhere in the
cafeteria and stop students from trying to enter periodically. The method then needs to
determine an optimal standing position. Different positions could lead to more serious side
effects, especially the possibility of obstructing the flow of individuals passing through the
cafeteria.16
Dynamic Simulation
In order to determine the most effective location for limiting the arrival rate, this study
used dynamic simulation to model the dining process to view the crowd density in different
areas. In order to determine an appropriate location.
Figure 10: Dynamic Simulation Preparation
We configured the elements based on the plan figure (Figure 1) of the cafeteria. As
shown in Figure 10, the orange areas represent walls, which are considered impassable for
pedestrians. Due to the narrow spacing between chairs and tables, it is assumed that pedestrians
cannot pass through the central seating area. The green line segments at the upper-left and
lower-left corners indicate the entrance and exit points of the cafeteria. The green arrows denote
the queueing direction, while the green dots represent the positions of the service windows.
Additionally, the green shaded area on the left side indicates a corridor for through-passage.
Subsequently, we employed agent-based modeling techniques to simulate pedestrian
behaviors within the cafeteria, utilizing the Pedestrian Library provided by AnyLogic
(AnyLogic Company, n.d.). This approach allows for detailed representation of individual
movements and interactions (Zhou, 2012). The previously computed time-varying arrival rate
function and service rate function were input. Pedestrian behavior was determined based on the
proportion functions P (t) and r. Each individual agent was removed from the simulation once
they had either completed dining within the cafeteria or exited the system, thereby marking the
end of their simulation lifecycle.17
Heatmap analysis (Kernel Density Estimation, KDE)
Figure 11: Heatmap visualization at the 200th second of the simulation
Heat maps are obtained based on Kernel Density Estimation (KDE), whose values do
not represent the absolute number of people, but rather the density of people in the vicinity per
unit area (Zhou, Tang, Wang, & Wang, 2013). Based on our feelings, a KDE value of more
than 4 in some areas indicates an extreme crowding of people in this area.
Through Figure 11 we can find that the crowded area is basically concentrated in a part
near the service windows, while the density in the corridor is relatively less. Therefore, we can
make a teacher or student volunteer to control the flow of people at the leftmost side of all
windows (Figure 12), which is where students must go through to enter the queueing area. This
could be done by releasing 14 - 16 people every 2 minutes for the first 10 minutes (maximum
arrival rate between 0.117 to 0.133).
Figure 12: Location of the flow control18
Discussion
In summary, validation of the optimization strategy shows that adding an additional
service window and limiting the average service time to 25 seconds can significantly reduce
waiting times during peak hours. Integrated with flow-control of releasing only 12–14 people
every two minutes. This approach effectively mitigates sudden surges in customer arrival,
thereby maintaining the system utilization rate, ρ, within the safe interval of 0.8–0.9 and
keeping the maximum average waiting time at approximately 50 seconds. Moreover, AnyLogic
dynamic simulation and heat-map analysis further reveal that crowding density peaks in the
queueing area rather than in the corridor, and that installing flow-control action alongside the
window can effectively alleviate the perception of congestion (Rossetti, 2021).
In addition to this, this study not only provides a quantifiable solution for the school,
but also an effective template for other school cafeterias and even broader situations with
similar characteristics and problems (e.g., varying arrival rates). Both the behavioral analysis
of crowds and the queueing theory modeling theory in this study can be applied to many similar
situations.
In contrast to existing research, the dynamic queueing-modeling framework presented
here is relatively rare in practical applications; most studies rely on steady-state models for
optimization, whereas this paper employs a dynamic G(t)/M /c model combined with
smoothing-spline functions for nonparametric estimation of service rates, thus making the
model more reflective of real-world conditions.
Nevertheless, there remain certain sources of inaccuracies and limitations in both model
design and data processing. First, the observational dataset spans only eight days and does not
account for holidays, special events, or other contingencies. This sample is not representative
enough. Second, student behavior is characterized in a simplified manner, only ”purchasing
meals” versus ”not purchasing meals” and indoor versus outdoor preference. And omitting
more complex actions such as ”jockeying” and ”reneging” (Sztrik, 2021). Furthermore, this
study does not explore in depth the psychological disparity between ”perceived congestion”
and ”actual congestion”, which is crucial for enhancing students’ dining experience. For
example, visually seeing crowding also affects one’s perception, even if there is no change in19
the density of the surrounding crowd (Yildirim & Akalin-Baskaya, 2007).
It should also be noted that time-varying queueing models may overlook cumulative
effects of customers within the system, potentially underestimating queue lengths or peak
delays (Whitt, 2006). Other error sources include approximations of the arrival process,
simplifications in service time distribution, and biases in manual observational records.
Notably, although service times were found to conform to an exponential distribution via the
Kolmogorov–Smirnov test. However, it has not been proved that there is no other more
consistent distribution that can explain it.
Future work may proceed in the following directions: (1) expand the sample size by
collecting at least 30 days of data to enhance result precision; (2) integrate surveys or
physiological indicators to quantify students’ perceived levels of crowding under different
optimized situations to conclude an experiment (3) extend the model to more school types (e.g.,
middle schools, universities) and cafeterias of different scales to evaluate its generalizability.
Conclusion
In this study, the effectiveness of queueing theory combined with behavioral simulation
in alleviating canteen congestion is demonstrated, and this study propose a dynamic-modeling
approach that uses time-varying data to capture evolving scenarios. Based on multi-day field
observations and a G(t)/M /c framework, we quantitatively reveal that system instability
during the lunch rush arises when arrival rates exceed service capacity. By adding an extra
service window, shortening service durations, and implementing flow-control measures, the
average waiting time was effectively reduced to under 50 seconds, providing a quantitative
foundation and technical roadmap for real-world campus deployment. AnyLogic dynamic
simulation and heat-map analyses also further pinpoint the optimal locations for flow control.
Although the sample period and behavioral model remain somewhat simplified, our
findings already furnish campus canteen managers with clear, quantitative guidance for
operational adjustments. Future work may extend the model’s generalizability and further
enhance and detailed pedestrian-flow management. At both theoretical and practical levels, this
study also offers a viable queueing-system analysis framework that can be applied to other
school cafeterias and even broader public-service settings.