Hospital system queue simulation

Conceptual Model of the System

The system to be simulated is a clinic that attends to patients sequentially. Patients arrive at the clinic following a distribution of inter-arrival times, wait in a waiting room if the doctor is busy, and then receive medical attention.

Variables and Parameters

• Inter-arrival Time ($T_a$): Distribution of the time between patient arrivals.
• Service Time ($T_s$): Distribution of the time it takes to attend to a patient.
• Number of Doctors ($N_m$): Number of available doctors to attend to patients.
• Waiting Room Capacity: Assumed to be unlimited for this model.
• Number of Patients per Simulation ($N_p$): 300 patients.
• Number of Simulations ($N_s$): 2036 simulations, calculated using Chebyshev's theorem.
• Queue Policies: The customer queue is managed under the FIFO principle, and the event queue is handled by the nearest time on the clock.

System Diagram with a Single Doctor

System Diagram with Multiple Doctors

Design of Relevant Events

The events considered in the simulation are:

• Arrival: A patient arrives at the clinic.
• Service Start: A patient begins to be attended by a doctor.
• Departure: A patient finishes their attention and leaves the clinic.

These events are managed through an event queue prioritized by the time they occur.

Statistical Analysis of Distributions

Goodness-of-fit tests were conducted to determine the distributions that best model the inter-arrival times and service times.

Inter-arrival Times

• The following distributions were tested: exponential, gamma, weibull_min, lognormal, beta, and triangular.
• The exponential distribution was the best fit for the inter-arrival times, according to the Kolmogorov-Smirnov test with a high p-value, and the null hypothesis was not rejected.

Distribution	KS Statistic	p-value	Decision
exponential	0.045565	0.5463812	Do not reject
gamma	0.051585	0.3886398	Do not reject
weibull_min	0.076667	0.05575857	Do not reject
lognormal	0.501763	$2.11 \times 10^{-70}$	Reject
beta	0.039920	0.7099192	Do not reject
triangular	0.384863	$1.72 \times 10^{-40}$	Reject

Table: Goodness-of-fit tests for inter-arrival times

Service Times

• The same distributions as for inter-arrival times were tested.
• The beta distribution was the best fit for service times, according to the Kolmogorov-Smirnov test with a high p-value, and the null hypothesis was not rejected.

Distribution	KS Statistic	p-value	Decision
exponential	0.318138	$1.68 \times 10^{-27}$	Reject
gamma	0.041910	0.651981	Do not reject
weibull_min	0.041420	0.666316	Do not reject
lognormal	0.042769	0.626883	Do not reject
beta	0.037464	0.779192	Do not reject
triangular	0.063318	0.172799	Do not reject

Table: Goodness-of-fit test results for service times

Calculation of the Optimal Number of Simulations

Using Chebyshev's theorem and considering an allowable error probability ($\alpha = 0.05$), the optimal number of simulations was calculated:

$$ m = \max(\text{Mean of } T_a, \text{Mean of } T_s) = \max(6.772, 10.0915) = 10.0915 $$

$$ N_s = \left( \frac{m^2}{\alpha} \right) = \left( \frac{10.0915^2}{0.05} \right) \approx 2036 $$

Results obtained:

• Mean of the inter-arrival time distribution: 6.772 minutes
• Mean of the service time distribution: 10.0915 minutes
• Optimal number of simulations: 2036

Simulation Results

System with a Single Doctor

2036 iterations of the system with one doctor and 300 patients per simulation were conducted. The average results were:

• Percentage of time the attention room is occupied: 99.60%
• Maximum time a patient can wait in the waiting room: 1003.24 minutes
• Average number of patients in the queue during peak hours: 49.92 patients
• Average total time a patient spends in the clinic: 512.84 minutes

These results indicate a high system saturation, with excessively long waiting and stay times. This makes sense since the average service time is almost double the average inter-arrival time, theoretically causing the queue to saturate infinitely over time.

Determination of the Required Number of Doctors

The minimum number of doctors required was sought so that no patient waits more than 10 minutes in 95% of cases.

Number of Doctors	95th Percentile of Waiting Time (min)	Requirement Met?
1	986.24	No
2	23.83	No
3	5.85	Yes

With 3 doctors, the established requirement is met.

Answers to the Raised Questions

What is the percentage of time the attention room is occupied?

In the system with a single doctor, the attention room is occupied 99.60% of the time, indicating high utilization.

What is the maximum time a patient can wait in the waiting room?

The maximum recorded waiting time was 1003.24 minutes, which is excessive and highlights the need to improve the system.

What is the average number of patients in the queue during peak hours?

The average number of patients in the queue during peak hours is 49.92 patients.

What is the average total time a patient spends in the clinic?

The average total time a patient spends in the clinic is 512.84 minutes.

How many additional doctors are needed to ensure that no patient waits more than 10 minutes in 95% of cases?

2 additional doctors are required, making a total of 3 doctors, to meet the maximum waiting time requirement.

Recommendations

• Increase the number of doctors to 3: This will significantly reduce waiting times and improve patient satisfaction.
• Optimize resource allocation: Continuously monitor demand and adjust resources accordingly.
• Improve doctors' productivity: Enhance efficiency to reduce service times.