The data shown below were obtained by time-lapse photography on a highway. Use regression analysis to fit these data to the Greenshields model and determine (a) the mean free speed, (b) the jam density, (c) the capacity, and (d) the speed at maximum flow.

Question

What will be an ideal response?

Answer 1

A linear regression analysis can be applied to the given data to estimate parameters in Greenshields’ model of traffic flow. Greenshields’ model (Equation 6.20) is:



A linear regression model takes the form y = a + bx; therefore, in this case, the

given data us and k correspond to y and x respectively. The linear regression

analysis can be performed using equations 6.21, 6.22, and 6.23, or using a

computer software spreadsheet package. Linear regression analysis yields values

of a = 62.8124 and b = –0.56845. Therefore,

a) Mean free flow speed, uf = a = 62.8 mi/h

b) Jam density, kj

In the regression model, b = uf / kj

b = 0.56845

kj = 62.8 / 0.56845 = 110.49

kj = 110 veh/mi

c) Capacity, qmax

Capacity occurs at maximum flow. State flow in terms of density.



Take the derivative and set equal to zero to maximize flow; solve for density.

density.

0 = 62.8 – 1.1368k

k = 55.25 when q = qmax

Solve for q

qmax = 62.8(55.25) – 0.5684(55.25)2

qmax = 1735 veh/h

d) Speed at maximum flow

Solve for mean speed using k when q = qmax