Question $4-$ Fixed bottleneck $[28]$

There are $3500$ vehicles per hour entering an arterial road with three lanes in each direction. The

fundamental diagram for this road can be modelled using a triangular flow$-$density relationship, having a

total capacity $($maximum flow$)$ of $4500$ vehicles per hour, a critical density of $100$ vehicles per kilometre,

and a jam density of $300$ vehicles per kilometre $($for $3$ lanes$).$ Assume that capacity of each lane is

identical.

Initially, the arterial carries traffic quite well until a car illegally parks, blocking one of the three lanes and

forcing traffic to merge into the other two lanes. A traffic queue quickly builds up behind the parked car.

Assuming that the parked car remains and after a period of $30$ minutes the arrival flow decreases from

$3500$ vehicles per hour to $1500$ vehicles per hour.

a$)$ Identify traffic states $($flow and density$)$ on the above arterial road $[6]$

b$)$ Calculate the shockwave speeds and plot the shockwaves on the fundamental diagram $[6]$

c$)$ Plot the time$-$space diagram and showing the shockwaves in this time$-$space diagram $[6]$

d$)$ Sketch some vehicle trajectories on the time$-$space diagram before and after the bottleneck $[4]$

e$)$ Calculate the queue of slow$-$moving vehicles reach in the $30$ minutes until the arrival flow decreases

$[3]$

f$)$ Calculate the time it takes the queue of slow$-$moving vehicles to dissipate after the arrival flow

decreases $[3]$