Question

A toll tunnel has decided to experiment with the use of a debit card for the collection of tolls. Initially, only one lane will be used. Cars are estimated to arrive at this experimental lane at the rate of 250 per hour. It will take exactly 12.0 seconds to verify the debit card.

How much time would you expect a customer to wait in line, pay with a debit card, and leave?

Answer 1

The expected wait time for a customer in line, paying with a debit card and leaving the toll tunnel, is approximately 3.45 minutes.

Step-by-step explanation:

To calculate the expected wait time for a customer in line, we need to use the formula for the $M/M/1$ queueing system. In this case, the arrival rate is 250 cars per hour and the service time is 12.0 seconds per car.

The formula for the expected wait time, also known as the average queueing time, is:

W = (λ / (μ - λ)) × S

Where λ is the arrival rate, μ is the service rate, and S is the service time. Plugging in the values, we get:

W = (250 / (4.167 - 250/3600)) × 12.0

Calculating this, we find that the expected wait time for a customer in line, paying with a debit card and leaving, is approximately 206.78 seconds or 3.45 minutes.