Question: Passengers arrive to a train platform according to a Poisson process at rate . As soon as /V passengers arrive, an empty train is sent

Passengers arrive to a train platform according to a Poisson process at rate . As soon as /V passengers arrive, an empty train is sent there to pick them up, but the train takes T units of time to get there. When it gets there it picks up all waiting passengers including new ones (if any ) that arrived during the T time units, and departs. There is a cost of $c per unit time per passenger waiting and a fixed cost $K each time the train departs with all the passengers. After departure, independently this system starts over and over again forever. YOU MUST SHOW YOUR WORK. (a) Find the long-run proportion of time that the platform is empty. (b) Find the long-run proportion of passengers that find the platform empty when they arrive. (c) Find the long-run rate at which trains depart. (d) Find the expected number of passengers who depart on a train. (e) Find the long-run cost rate for this system

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