Question

The Quick Snap photo machine at the Lemon County bus station takes snapshots in exactly 80 seconds. Customers arrive at the machine according to a Poisson distribution at the mean rate of 15 per hour. On the basis of 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 = 15 per hour
Exact service rate, μ = 1 in 80 seconds = 45 per hour

Since the service rate is deterministic, this is an example of an M/D/1 queue.

(a)

The average number of customers waiting, L_q = λ² / {2μ.(μ - λ)} = (15^2) / (2*45*(45 - 15)) = 0.0833

(b)

The average waiting time, W_q = L_q / λ = 0.0833 / 15 = 0.00556 hours = 20 seconds

So, the average time in the system, W_s = W_q + service time = 20+80 = 100 seconds

(c)

Probability of an arriving customer waits = λ/μ = 15/45 = 0.333