Traffic Flow Model

A study of freeway flow at a particular site has resulted in a calibrated speed-density relationship, as follows:

U_s= 57.5(1-0.008k)

From this relationship:

1. Find the free-flow speed and jam density
2. Derive the equations describing flow versus speed and flow versus density.
3. Determine the capacity of the site mathematically

 [Solution Shown Below]

A.) To solve for free-flow speed and jam density:

u_s = 57.5 –0.46k. Notice that this equation is linear with respect to space mean speed and density and is of the form of Greenshield’s equation.

Greenshield’s equation: u_s= u_f- (u_f/k_j)k

Free flow speed u_f = 57.5 MPH

To calculate jam density: u_f/k_j = 0.46 gives k_j = 125 vpm

B.) To derive the equations for flow as a function of density:

q= u_sk

q = 57.5k-0.46k² vph gives flow as a function of density ( note that it is a quadratic in k)

To derive flow as a function of speed:

0.46k=57.5-u_s

k=(57.5-u_s)/0.46 = 125-(u_s/0.46)

q = u_s(125-(u_s/0.46)) = 125u_s- u_s²/0.46 vph ( note that it is a quadratic in u_s)

C.) To determine the capacity of the site:

Need to determine the maximum flow:

dq/dk = 57.5 – 0.46(2)k =0

57.5=0.46(2)k

k = 57.5/(0.46(2)) = 62.5 veh per mile = k_m=density at maximum flow

q = 57.5k-0.46k²

q=57.5(62.5) –0.46(62.5)²

q=3593.75 –1796.875

q = 1796.875 veh/hour = q_m

speed at maxium flow = u_m = 57.5 –0.46(62.5) =28.75 mph