Question

The arrival of patients in a clinic (with only 1 doctor) follows a Poisson process with a level of 30 patients per hour. The clinic has a waiting room that can accommodate no more than 14 people. The doctor's service time at the clinic follows an exponential distribution with an average of 3 minutes per patient.
a. Determine the opportunity that a patient who comes doesn't need to wait.
b. Determine the opportunity that a patient who comes will find an empty seat in the clinic.
c. Determine the average time spent by a patient at the clinic.

Answer 1

Arrival rate = 30 patients per hour = λ

Service time = 3 minutes = µ

Number of patients served in an hour = 60/3 = 20 patients

Average time a customer spends waiting in line for service

W_q = λ/µ(µ - λ)

Average time spent by a patient in the clinic = W_q + 1/µ

Probability that their are zero customers in the clinic = 1 - λ/µ

λ>µ

If the arrival rate is greater than or equal to the service rate, there is no stationary distribution and the queue will grow without bound.

Arrival rate is greater than service rate, so it's applied in formula the answer will come negative, which is not possible that time and number of customers comes negative.