Question

Patients arrive randomly at an eye care clinic for 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 wait in a waiting room until the optometrist is ready to see them. How many patients, on the average, will be in the waiting room?

Answer 1

Using queueing theory, the formulas for average number of patients in the system (L) and in the waiting room (Lq) are applied with the given arrival and service rates, resulting in an average of 0.9 patients in the waiting room.

Step-by-step explanation:

To calculate the average number of patients in the waiting room of the eye care clinic, we use concepts from queueing theory. In particular, we will use the Poisson distribution and exponential distribution to understand the arrival and service processes.

Given that the average arrival rate (λ) is 12 patients per hour and the average service rate (μ) is 20 patients per hour, we can determine the average number of patients in the system (L) using the formula:

L = λ / (μ - λ)

Plugging in the values, we get:

L = 12 / (20 - 12)

L = 12 / 8

L = 1.5

However, this number includes the patient being served. To find the average number of patients in the waiting room (Lq), we use the formula:

Lq = λ² / μ (μ - λ)

So, we calculate it as:

Lq = 12² / 20 (20 - 12)

Lq = 144 / 160

Lq = 0.9

Therefore, on average, there will be 0.9 patients in the waiting room.