OUESTION$\backslash$#$4(7$ POINTS$)$

Suppose that two shuttle buses operate between the Waterford Shopping Center in Southeast Orange County and the UCF campus. One bus can carry $20$ people and has a round trip time of $8$ minutes $($including loading and unloading passengers$)$; the other can carry $25$ people but has a round time of $10$ minutes. There is always a queue of people waiting at the Waterford Shopping Center bus stop, but people arrive at the UCF campus bus stop at a steady rate of $2$ per minute.

If neither bus is idle at any time, determine:

$(4\mathrm{.}1)$ Hourly flow of people from Waterford Shopping Center to UCF campus.

$(4\mathrm{.}2)$ Hourly flow of people from UCF campus to Waterford Shopping Center.

$(4\mathrm{.}3)$ Hourly flow of buses from Waterford Shopping Center to UCF campus.

$(4\mathrm{.}4)$ Hourly flow of buses from UCF campus to Waterford Shopping Center.

$(4\mathrm{.}5)$ The average number of passengers per bus from UCF campus to Waterford Shopping Center.

$(4\mathrm{.}6)$ If one insisted on dispatching these buses from Waterford Shopping Center to UCF campus at equal headways $($by asking the faster bus driver to wait two extra minutes in every round trip$),$ what is the maximum number of people that could be carried per hour from Waterford Shopping Center to UCF campus?

$(4\mathrm{.}7)$ Would you expect the strategy in $(4\mathrm{.}6)$ above to reduce the average waiting time of passengers going from UCF campus to Waterford Shopping Center? Justify your answer by calculating waiting times with a numerical example using data in this problem. State all your assumptions clearly.