Question

A ferry loads cars for delivery across a river and must have a full ferry load of 10 cars. Loading is instantaneous and the round-trip time is an exponential random variable with mean of 15 min. The cars arrive at the shore as a Poisson process with a mean of 30/h. On average, how many autos are waiting on the shore at any instant for a ferry?

Answer 1

The average number of autos waiting on the shore at any instant for a ferry is 8 autos.

Step-by-step explanation:

To find the average number of autos waiting on the shore at any instant for a ferry, we need to calculate the average arrivals of cars and the average time it takes for the ferry to complete a round-trip.

1. First, we calculate the average time between car arrivals using the Poisson process. The mean time between car arrivals is 1/(mean rate of arrival) = 1/(30/h) = 1/0.5 = 2 hours.
2. Next, we calculate the average round-trip time for the ferry, which is an exponential random variable with a mean of 15 minutes. Therefore, the average time for a round-trip is 15 minutes.
3. Finally, we divide the average time between car arrivals by the average round-trip time to get the average number of autos waiting on the shore at any instant for a ferry: 2 hours / 0.25 hours (15 minutes) = 8 autos.