Customers arrive at Rent a Wreck and hope to score an inexpensive and unsightly vehicle to use for what typically is a short period of time. About twenty customers show up in any hour of the day and it takes the customer service representative about two minutes to photocopy their driver's license and insurance card and take an impression of a credit card. Assume that the service and arrival rates are governed by the Poisson distribution; how long does it take for the average customer to clear the system?

Question

Quizplus · Accepted Answer

Since the arrival rate and service rate are Poisson distributed, we can use the M/M/1 queuing model. 
λ = 20 customers/hour 
μ = 30 customers/hour (since each customer takes 2 minutes, µ = 1/2 customers/minute; there are 60 minutes in an hour, so µ = 30 customers/hour) 
Using the formula for the expected waiting time in an M/M/1 queuing model, we get: 
W = (1/µ) / (1 - λ/µ) 
W = (1/30) / (1 - 20/30) 
W = (1/30) / (1/3) 
W = 3 minutes 
Then, we add the average service time of 2 minutes to get the average time a customer spends in the system: 
3 minutes + 2 minutes = 5 minutes 
Therefore, the average time for a customer to clear the system is 5 minutes. However, since the question asks about how long it takes for the average customer to clear the system, we need to divide by the arrival rate: 
5 minutes / 20 customers/hour = 0.25 hours or 15 minutes 
Finally, we round up to get the best choice: 6 minutes.

Customers Arrive at Rent a Wreck and Hope to Score