In English, the given statement is:

$"$On average, $5$ patients per hour $($Poisson distribution$)$ arrive at the campus walk$-$in health clinic without appointments. The average time to serve a patient is $27$ minutes $($exponential distribution$).$ The clinic has $2$ available doctors to attend to patients. When the waiting room is full $(15$ seats$),$ the clinic redirects new patients to the nearby hospital's emergency room.

a$)$ Calculate:

$-$ Doctor utilization

$-$ Average number of patients in the queue

$-$ Average time a patient spends in the queue

$-$ Probability that a patient arriving will be redirected to the emergency room

$-$ Probability that there are more than $5$ patients in the queue when a patient arrives.

b$)$ The goal is to reduce the average time spent in the waiting room to below $40$ minutes. There are $3$ options:

$1-$ Reduce the average service time to $20$ minutes per patient by hiring a nurse practitioner $($$$100/$hour$).$

$2-$ Reduce the size of the waiting room. Charge $$10$ per removed seat.

$3-$ Add another doctor $($$$200/$hour$).$

Additionally, the clinic incurs an opportunity cost of $$200$ for each patient redirected to the emergency room.

Recalculate the values from part a$)$ and determine the total cost for each option $($scenarios $1,2,3).$ Based on the costs, which option would you recommend?"