Problem $1[20$ points$]$

Consider a $3-$lame homogeneous section of freeway, meaning all three lanes have identifcal traffic flow properties. The jam spacing in one lane is sj $=30$ ft$/$veh$,$ the free$-$flow speed is vf $=55$ mph$,$ and the minimum vehicle headway $($i$.$e$.,$ at capacity$)$ is hc $=1\mathrm{.}8$ sec$/$veh$.$

On a particular day when traffic is flowing at a rate of $1800$ veh$/$hr per lane, and incident occurs on the roadside that distracts drivers and causes them to slow down to look. This phenomenon is called rubbernecking. The rubbernecking causes the density of traffic to increase in the vicinity of the incident to increase to $100$ veh$/$mile per lane. After $15$ minutes, the incident is cleared up and the traffic begins to recover to normal conditions.

a$)$ Draw the fundamental diagram for this section of highway, assuming that it is triangular. Identify and label the relevant traffic states.

b$)$ Draw a time$-$space diagram showing the interfaces or shockwaves between traffic states, and sketch some of the trajectories. Label the traffic states on the time$-$space diagram corresponding to the states you identified in part $($a$).$ Calculate and label the speeds of the interfaces or shockwaves.

c$)$ How long does it take for the entire section of road to resume normal traffic conditions?

d$)$ What is the maximum extent $($i$.$e$.,$ the physical length$)$ of the queue from the incident?