Two routes connect an origin-destination pair with performance functions t1 = 6 + 4x1 and t2 = 2 + 0.5x2 2 (with t's in minutes and x's in thousands of vehicles per hour). The origin-destination demand is 4000 veh/h at a travel time of 2 minutes, but for each additional minute beyond these 2 minutes, 100 fewer vehicles depart. Determine user equilibrium route flows and total vehicle travel time.

i) 4 veh/h

ii) 3,400 veh/h

Explanation:

The equilibrium route flow, T = t2 - t1

Hence, T = 0.5x² + 2 - (6 + 4x)

T = 0.5x² + 2 - 6 - 4x = 0.5x² - 4x - 4

dT/dx = x - 4

dT/dx = o

x - 4 = 0

∴ = 4

The user equilibrium route flow = 4 veh/h

ii) When x = 4 veh/h, T = 8 - 16 - 4 = - 12 mins

Hence, the additional minutes beyond 2 minutes is 12 minutes

∴ 12min/2min X 100 vehicles = 600 vehicles

The total vehicles travel time = (4,000 - 600) veh/h = 3,400 veh/h