Problem $2$: For the situation described in Problem $1,$ suppose that the freeway authority

proposes a new tolling scheme to mitigate congestion. The new tolling scheme will charge a

higher toll rate of $$3$ per passing to vehicles that travel in peak hours between $8$:$00$ AM$-$

$10$:$00$AM$.$ For vehicles that travel during non$-$peak hours $($i$.$e$.,$ before $8$:$00$ AM or after

$10$:$00$AM$),$ the toll rate is the same as the existing toll rate of $$2$ per passing. If the new toll

scheme was implemented, some peak$-$hour travelers would be expected to switch to non$-$peak

hours to take advantage of the lower toll rate. The new arrival rates are predicted as follows.

Here, the capacity of the freeway is still $5,000ve\frac{h}{h}.$

Table $2$: Arrival patterns given the proposed time$-$dependent toll rates $($$$3$ per passing for $8$:$00-$

$10$:$00$AM; $$2$ per passing for other time$)$

a$)$ Draw the cumulative plots of arrivals and departures, separately, to demonstrate the

predicted traffic flow patterns due to the impact of the higher toll rate charged to peak $-$

hour users.

b$)$ For the scenario of toll pricing, what is the total amount of delay experienced by the

affected vehicles?

c$)$ For the scenario of toll pricing, what is the average delay per vehicle?

d$)$ Discuss the impact of the time$-$dependent tolling scheme on traffic congestion based on

your analysis.