*48. In Example 7.20, let denote the proportion of passengers that wait less than x for...

*48. In Example 7.20, let π denote the proportion of passengers that wait less than x for a bus to arrive. That is, with equal to the waiting time of passenger i, if we define

(a) With N equal to the number of passengers that get on the bus, use renewal reward process theory to argue that Image

(b) With T equal to the time between successive buses, determine Image.

(c) Show that Image.

(d) Show that

(e) Using that is the proportion of time that the excess of a renewal process with interarrival times distributed according to T is less than x, relate the result of

(d) to the PASTA principle that “Poisson arrivals see the system as it averages over time”.