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In a hospital emergency department, patients arrive at an average rate of $15$ patients per hour, following a Poisson process. The department is divided into two sections:

Triage Section $($M$/$M$/1$ Queue$)$: Patients are initially assessed by a single triage nurse who can process patients at a rate of $20$ patients per hour, with service

times following an exponential distribution. After triage assessment, patients proceed to the treatment section.

Treatment Section $($M$/$M$/2$ Queue$)$: After triage, patients enter the treatment section, where two doctors are available. The treatment section operates as an M$/$M$/2$ queue, where each doctor can treat patients at an average rate of $12$ patients per hour, with service times following an exponential distribution.

After treatment:

$85\%$ of patients are discharged.

$15\%$ of patients require further evaluation and are sent back to the triage section for reassessment before returning for additional treatment.

Answer the following questions:

Triage Section Analysis:

Calculate the effective arrival rate at the triage section, considering that $15\%$ of patients return to the triage section after their initial treatment.

Determine the utilization rate $()$ for the triage nurse based on this adjusted arrival rate.

Calculate the average number of patients $($ L $)$ in the triage section, including both new arrivals and returning patients.

Find the average waiting time $(Wq)$ for a patient in the triage queue before being assessed or reassessed.

Treatment Section Analysis:

Calculate the effective arrival rate at the treatment section, taking into account both new patients from triage and those returning after reassessment.

Determine the utilization rate $()$ for the treatment section with the two doctors.

Calculate the average number of patients $($L$)$ in the treatment section, including both new patients and those returning for further treatment.

Find the average waiting time $(Wq)$ in the treatment queue before a patient is seen by a doctor.

Scenario Planning:

During peak hours, the patient arrival rate may increase to $20$ patients per hour. Recalculate the metrics $(,L,$ and Wq $)$ for both the triage and

treatment sections under this increased arrival rate, considering patients returning for reassessment and treatment. Discuss the impact on wait times and whether additional staffing would be required.

System Recommendations:

If the calculated utilization for either section exceeds $90\%$ during peak times, recommend a possible improvement $($e$.$g$.,$ adding an additional triage nurse or doctor$).$ Discuss how this adjustment would impact patient wait times, the handling of returning patients, and overall system flow.