Problem $2$ on $\frac{D}{D}\frac{?}{1}$ queuing $(40$ points$)$: Vehicles arrive at a single toll booth beginning at $8$:$00$ A$.$M$.$ They arrive and depart according to a uniform deterministic distribution. However, the toll booth does not open until $8$:$10$ A$.$M$.$ The average arrival rate is $8ve\frac{h}{m}in$ and the average departure rate is $10ve\frac{h}{m}in.$

Please construct a cumulative arrival and departure flow count diagram to visualize the scenario. On the diagram, label the following:

$CA(t)$: Cumulative Arrivals up to time $t$

$,CD(t)$ : Cumulative Departures up to time $t$

Indicate the peak queue length $q_{\text{m}\text{a}\text{x}}$

Point A: The queue begins to form

Point B: Service commences

Point C: Time at which the initial queue is cleared, i$.$e$.Tc$

Assuming $\frac{D}{D}\frac{?}{1}$ queuing, when does the initial queue clear and what are the total delay, the average delay per vehicle, longest queue length $($in vehicles$),$ average queue length, and the wait time of the $100$ th vehicle to arrive $($assuming first$-$in$-$first$-$out$)?$