Consider a ramp with two lanes: one general purpose $($GP$)$ lane and one

high$-$occupancy vehicle $($HOV$)$ lane. Vehicles on the GP lane may wait at the ramp meter, while vehicles on the HOV lane enter without having to stop. $20\%$ of the vehicles are HOVs with $2\mathrm{.}4$ passengers $($incl$.$ the driver$)$ on average, while the other vehicles have single occupancy $($i$.$e$.,$ only one driver$).$ The demand at the ramp is $500$ veh$/$hr during the entire studied period except for $1$ hour of peak demand of $1000$ veh$/$hr$.$ The mainline demand is $3600$ veh$/$hr during the off$-$peak hours and $5200$ veh$/$hr during the peak hours. The mainline capacity is $6000$ veh$/$hr$.$

$2\mathrm{.}1.$ Determine the optimal ramp flow for the GP lane during the peak hour.

$2\mathrm{.}2.$ Determine the total delay for the vehicles that use the GP lane. What is the time savings in terms of total passenger delay $($i$.$e$.,$ passenger hours travelled$)$ compared to a scenario where all vehicles are treated the same?

$2\mathrm{.}3.$ Assume that each ramp lane has a triangular fundamental diagram with a free$-$flow speed of $80$ km$/$hr$,$ a capacity of $2000$ veh$/$hr$,$ and a jam density of $150$ veh$/$km$.$ Determine the minimum length of the ramp such that there is no spillback $.$