$3.$ Suppose there are two shuttle buses operating between $($a bus stop at$)$ a remote parking facility and the "front gate" of the U$.$C$.$ One bus can carry $20$ people and has a round trip time of eight minutes $($including passenger loading $\backslash$& unloading times$),$ the other can carry $25$ people but has a round trip time of $10$ minutes. There is always a queue of people waiting at the bus stop in the remote parking area. People wishing to board a shuttle and return to their cars arrive to the U$.$C$.\text{'}$s front gate at a steady rate of $2$ per minute.

If neither bus is idle at any time, determine the flow

a$)$ of bus passengers arriving to the front gate from the remote parking lot,

b$)$ of bus passengers arriving to the remote parking lot from the front gate,

c$)$ of buses from the remote parking lot to the front gate,

d$)$ of buses from the front gate to the remote parking lot,

e$)$ the average number of passengers per bus from the front gate to the remote parking.

If one insisted on dispatching these shuttle buses at equal headways, what is the maximum number of people that could be carried per hour from the remote parking lot to the front gate?

Would the $($above$)$ strategy of equal headways reduce the average waiting time of passengers going from the front gate to the remote parking?