Norman W. Garrick

Traffic Stream Flow Equations

Norman W. Garrick

Basic Stream Flow Parameters

Three types of parameters

1. Spacing and Concentration

2. Headway and Flow

3. Speed

Norman W. Garrick

Spacing is the distance between vehicles, s

Concentration (or density) is the number of vehicles

per unit length, k

Spacing and Concentration

s2 s3 s4

D

s(ave) = 1 / k

s1

Norman W. Garrick

Headway is the time between vehicles passing a

specific location, h

Flow is the amount of vehicles passing a specific

location in a specific time, q

Headway and Flow

h(ave) = 1 / q

measurement point

Norman W. Garrick

TMS and SMS

Time Mean Speed - is based on the spot speed of the vehicles passing a specific point

Space Mean Speed - is based on the average time it takes vehicles to cover a certain specified distance

Time Mean Speed, ut = 1/n ∑ vi

Space mean speed, us = 1 / {1/n ∑ 1/vi }

Norman W. Garrick

Relationship between Space and Time Parameters

What is the concentration?k = 100 vehicles/mile

What is the space?s = 1/k = 1/100 miles = 52.8 feet

If the average speed = 10 milesWhat is the flow (q)?

Flow is total number of vehicles passing any point Flow = 10*100 = 1000 vehicles/hr

In other words, q = uk

100 vehicles

D = 1 mile

Norman W. Garrick

Representing Traffic Flow

The fundamental equation of traffic stream flow is

q = ukThis equation is used in conjunction with the Traffic Flow Curves (next

slide) in an attempt to characterize traffic flow on a macroscopic level.

However, it should be understood that these analytical representation of traffic flow does not come close to accurately depicting the real life traffic which is a complex and non-linear phenomena.

The complexity derive from the fact that we have a large number of

vehicles interacting in a way that does not follow the laws of mechanics due to the reaction of the HUMAN driver

Norman W. Garrick

Traffic Flow Curves

u

k

Norman W. Garrick

Traffic Flow Curves

u

k

Norman W. Garrick

Traffic Flow Curves

u

k

u

q

Norman W. Garrick

Traffic Flow Curves

u

k

u

q

One important point to note, FLOW increase with SPEED but only up to a point

Why?

Norman W. Garrick

Traffic Flow Curves

u

k

u

q

k

q

Norman W. Garrick

What is the Maximum Theoretical Flow on a Highway?

If we have observed Density and Speed we could estimate maximum flow

The relationship between SPACE and SPEED depends on the ‘safety regime’

This safety regime is hard to characterize and depends on factors such as weather, age of drivers and other cultural factors

The figure in the next slide is considered to give a SPACE/SPEED relationship that comes close to reality for American driving conditions

Norman W. Garrick

Calculating Maximum FlowSpacing versus Speed

0

100

200

300

400

500

600

0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0

Speed, mph

Sp

aci

ng

, fe

et

Curve A from Figure 3.2.2 in Text

Spa

cing

, ft

Norman W. Garrick

Calculating Maximum FlowSpeed versus Concentration (u-k)

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

0 50 100 150 200 250

Concentration (k), vehicles/ mile

Speed, m

ph

k = 1 / s(ave) u

k

Norman W. Garrick

Calculating Maximum FlowFlow versus Speed (u-q)

0.0

20.0

40.0

60.0

80.0

100.0

120.0

140.0

0 500 1000 1500 2000 2500

Flow (q), vehicles/ hr

Speed (u

), m

ph

In theory qmax occurs at speed of about 40 mph

u

q

Norman W. Garrick

Calculating Maximum FlowFlow versus Concentration (q-k)

0

500

1000

1500

2000

2500

0 50 100 150 200 250

Concentration (k), vehicles/ mile

Flo

w (

q), veh

icle

s/h

r

q = ukqmax = 2000 veh/hr

k

q

Norman W. GarrickNorman W. Garrick

Traffic Flow CurvesMaximum Flow, Jam Concentration, Freeflow Speed

u

k

u

q

k

qqmax

qmaxkj

uf

kj - jam concentrationu = 0, k = kj

qmax - maximum flow

uf - free flow speedk = 0, u = uf