Question

The photo booth at the Apple County fair takes snapshots in exactly 90 seconds (constant rate). Customers arrive at the machine according to a Poisson distribution at the mean rate of 20 per hour. Using this information, determine the following:

a. the average number of customers waiting to use the photo machine

b. the average time a customer spends in the system

c. the probability an arriving customer must wait for service.

Answer 1

Average arrival rate, λ = 20 per hr.
Constant service rate, μ = 1 in 90 seconds = (3600/90) = 40 per hr.

(a)

The average number of customers waiting, L_q = λ² / {2μ.(μ - λ)} = (20^2) / (2*40*(40 - 20)) = 0.25

(b)

The average number of customers in the system, L_s = L_q + λ/μ = 0.25 + 20/40 = 0.75

The average time a customer spends in the system, W_s = L_s / λ = 0.75 / 20 = 0.0375 hrs. = 2.25 minutes

(c)

Probability that an arriving customer waits = λ/μ = 20/40 = 0.50