$(4$ points total$)$ Deterministic Queuing

Vehicles arrive at a single toll booth beginning at $8$:$00$ A$.$M$.$ They arrive and depart according

to a uniform deterministic distribution. The toll booth does not open until $8$:$10$ A$.$M$.$ The arrival

rate is $8ve\frac{h}{m}in,$ and the departure rate is $10ve\frac{h}{m}in.$ Answer the following questions:

a$.(1\mathrm{.}5$ points$)$ Sketch the queuing diagram showing the number of arrivals and the number

of departures. Plot time $($in minutes$)$ on the x $-$axis and vehicles on the y $-$axis.

Note: you only need to show the morning period when a queue forms.

b$.(0\mathrm{.}5$ points$)$ When does the initial queue clear?

c$.(0\mathrm{.}5$ point$)$ What is the total delay? Please express your answer in vehicle$-$minutes.

d$.(0\mathrm{.}5$ points$)$ What is the average delay per vehicle $($in minutes$)?$

e$.(0\mathrm{.}5$ points$)$ What is the longest queue length $($in vehicles$)?$

f$.(0\mathrm{.}5$ point$)$ What is the wait time $($in minutes$)$ of the $100$ th vehicle to arrive $($assuming

first$-$in$-$first$-$out$)?$