8. A facility produces items according to a Poisson process with rate . However, it has shelf...

8. A facility produces items according to a Poisson process with rate λ. However, it has shelf space for only k items and so it shuts down production whenever k items are present. Customers arrive at the facility according to a Poisson process with rate μ. Each customer wants one item and will immediately depart either with the item or empty handed if there is no item available.

(a) Find the proportion of customers that go away empty handed.

(b) Find the average time that an item is on the shelf.

(c) Find the average number of items on the shelf.