If speed and density observations give the data shown below, develop an appropriate expression for speed versus density for this highway, and determine the density at which the maximum volume will occur as well as the value of the maximum volume. Also plot speed versus density and volume versus speed for both the expression developed and the data shown. Comment on the differences between the two sets of curves.

Results of traffic flow studies on a highway indicate that the flow-density

relationship can be described by the expression:

What will be an ideal response?

Note that the given traffic flow relationship,

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, while a = uf and b = uf/kj.

The linear regression analysis can be performed using Equations 6.28, 6.29, and

6.30, or using a computer software spreadsheet package. Linear regression

analysis yields values of a = 63.949 and b = –0.4941. Therefore,

a) mean free flow speed, uf

uf = a = 63.9 mi/h

jam density, kj

uf / kj = b = –0.4941

kj = 63.9 / 0.4941 = 129.33

jam density, kj = 129.33 veh/mi

b) density at maximum volume

State flow in terms of density.

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

density.

0 = 63.9 – 0.9882k

k = 64.66 veh/mi when q = qmax

Solve for q

q = 63.9(64.66) – 0.4941(64.66)2

q = 2066 veh/hr

(a) maximum volume, qmax = 2066 veh/h

(b) density at maximum volume, k = 65 veh/mi

(c) Plot speed vs. density for data points and equation

(d) Comment on the differences between the two curves.

The equation used to describe the speed-density relationship is linear, thereby

yield a straight-line relationship as shown above. The data follow an

approximately linear relationship, the extent of which could be quantified using

the R2 value that would be obtained through regression analysis or by calculating

the sum of squared errors (SSE) between the equation and the data points.