Exercise $4\mathrm{.}14$ Patients arrive at an emergency room where they are treated and then depart. Arrivals are exponentially distributed with a mean time between arrivals of $0\mathrm{.}3$ hours. Upon arrival, patients are assigned a rating of $1$ to $5,$ depending on the severity of their ailments. Patients in Category $1$ are the most severe, and they are immediately sent to a bed where they await medical attention. All other patients must first wait in the receiving room until a basic registration form and medical record are completed. They then proceed to a bed.

The emergency room has three beds, one registration nurse, and two doctors. In all cases, the priority for allocating these resources is based on the severity of the ailment. Hint: Read the help for the QUEUE module and rank the queue by the severity attribute. The registration time for patients in Categories $2$ through $5$ is Uniform $(0\mathrm{.}1,0\mathrm{.}2)$ hours. The treatment time for all patients is triangularly distributed with the minimum, most likely, and maximum values differing according to the patient$$s category. The distribution of patients by category and the corresponding minimum, most likely, and maximum treatment times are summarized below.

Category $12345$

Percent $68183335$

Minimum $0\mathrm{.}80\mathrm{.}70\mathrm{.}40\mathrm{.}20\mathrm{.}1$

Most Likely $1\mathrm{.}20\mathrm{.}950\mathrm{.}60\mathrm{.}450\mathrm{.}35$

Maximum $1\mathrm{.}61\mathrm{.}10\mathrm{.}750\mathrm{.}60\mathrm{.}45$

The required responses for this simulation include:

Average number of patients waiting for registration

Utilization of beds

System time of each type of patient and overall $($across patient types$)$

Using a run length of $30$ days, develop a model to estimate the required responses. Report the responses based on estimating the system time of a patient regardless of type based on $50$ replications.