Consider a ferry that carries cars across a river. The ferry holds an integer number k of cars and departs the dock when full. At that time, a new ferry immediately appears and begins loading newly arriving cars ad infinitum. The ferry business has been good, but customers complain about the long wait for the ferry to become full.

(a) Assume that cars arrive according to a renewal process. The IID interarrival times have mean X, variance σ 2, and moment generating function gX (r). Does the sequence of departure times of the ferries form a renewal process? Explain carefully.

(b) Find the expected time that a customer waits, starting from its arrival at the ferry terminal and ending at the departure of its ferry. Note 1: Part of the problem here is to give a reasonable definition of the expected customer waiting time. Note 2: It might be useful to consider k = 1 and k = 2 first.

(c) Is there a 'slow truck' phenomenon (a dependence on E rX2l) here? Give an intuitive explanation. Hint: Look at k = 1 and k = 2 again.

(d) In an effort to decrease waiting, the ferry managers institute a policy where no customer ever has to wait more than one hour. Thus, the first customer to arrive after a ferry departure waits for either one hour or the time at which the ferry is full, whichever comes first, and then the ferry leaves and a new ferry starts to accumulate new customers. Does the sequence of ferry departures form a renewal process under this new system?

Does the sequence of times at which each successive empty ferry is entered by its first customer form a renewal process? You can assume here that t = 0 is the time of the first arrival to the first ferry. Explain carefully.

(e) Give an expression for the expected waiting time of the first new customer to enter an empty ferry under this new strategy.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.