Question $5(20$ points$)$: A toll booth on a turnpike is open from $8$:$00$ A$.$M$.$ to $12$ midnight. Vehicles start arriving at $7$:$45$ A$.$M$.$ at a uniform deterministic rate of five per minute until $8$:$15$ A$.$M$.$ and from then on at two per minute. If vehicles are processed at a uniform deterministic rate of five per minute, please $(1)$ draw a diagram showing the queuing process, $(2)$ determine when the queue will dissipate, $(3)$ the total delay, $(4)$ the maximum queue length $($in vehicles$),$ and $(5)$ the longest delay under First$-$in$-$First$-$out.