Question

Patients arrive randomly at an eye care clinic for an eye exam. Suppose that there is only one optometrist. The time required for the exam varies from patient to patient. Arrivals have been found to follow the Poisson process (i.e., exponentially distributed inter-arrival times), and the service times follow the exponential distribution. The average arrival rate is 12 patients per hour, and the average service rate is 20 patients per hour. A patient waits in a waiting room until the optometrist is ready to see them. What is the average number of patients, on average, that will be in the waiting room?

A) 3
B) 0.9
C) 36
D) None of the above

Answer 1

Using the M/M/1 queueing theory formulas, the average number of patients in the waiting room is found to be 0.9. This is calculated by plugging the given arrival rate (λ) and service rate (μ) into the formula for Lq: Lq = λ^2 / (μ (μ - λ)).

Step-by-step explanation:

The question involves a system where the arrivals follow a Poisson process, and the service times are exponentially distributed. Given that the average arrival rate (λ) is 12 patients per hour and the average service rate (μ) is 20 patients per hour. We are asked to find the average number of patients in the waiting room.

To solve this question, we can use the formula for the average number of customers in the waiting queue (Lq) for an M/M/1 queue: Lq = λ^2 / (μ (μ - λ)).

Plugging in the values λ = 12 and μ = 20:

Lq = (12^2) / (20*(20 - 12))
= 144 / (20*8)
= 144 / 160
= 0.9

So, the average number of patients in the waiting room is 0.9, which corresponds to option B).

Answer 2

Final answer:

Using the formula Lq = (λ^2) / (μ(μ - λ)), where λ is 12 and μ is 20, the average number of patients in the waiting room is calculated to be 0.9. Hence, the answer is B) 0.9.

Step-by-step explanation:

The average number of patients waiting in the room can be determined using the formula Lq = (λ^2) / (μ(μ - λ)), where λ is the average arrival rate and μ is the average service rate. In this case, λ = 12 patients per hour and μ = 20 patients per hour.

Plugging the values into the formula gives us Lq = (12^2) / (20(20 - 12)), which simplifies to Lq = 144 / (20 × 8) = 144 / 160 = 0.9.

Therefore, the correct answer is B) 0.9, which represents the average number of patients in the waiting room at any given time.