1. Macroscopic Density Characteristics
Traffic density as a macroscopic characteristic of traffic flow is used to assess traffic performance from the point of view of users and system operators.
Density is also employed as the primary control variable in freeway control and surveillance systems.
Measuring density was extremely difficult, and this inhibited the use of density as a primary traffic characteristic until the development of presence detectors in the early 1960s.

2. Why Density Matters?

3. Example of Density-Based Freeway Control
ALINEA Algorithm, a ramp metering algorithm
Maintaining the maximum flow at the downstream merge area
Successful applications: Paris
Formula
r(k) = r(k-1) + KR[Oc –Oout(k)]
k = time step
r(k) = metering rate in time step k
r(k-1) = metering rate in time step k-1
KR = an constant
Oout(k) = current measured occupancy
Oc = desired occupancy

4. Density Definitions
Density is defined as the number of vehicles occupying a length of roadway.
Density can be visualized by considering an aerial photograph of a section of highway, and to count the number of vehicles in a single lane having a length of one (1) mile.
Traffic densities vary from 0 to values representing vehicles that are bumper to bumper and stopped. This upper limit, defined as the jam density, is in the range of vpmpl.
Corresponding to 21 to 29 ft spacing, (roughly 20 to 30ft)

5. Density, Capacity, & Congestion
Optimum density is defined as the density that exists when the traffic flowing at capacity.
Optimum density occurs at levels of vpmpl, with an average spacing of ft/veh.
When density exceeds optimum density, traffic flow is in congested conditions.
55 to 85 mph at capacity.

6. Density Regions Density scale can be subdivided into three regions.
Region 1: Values from 0-42 vpmpl denote flow conditions in which traffic demands are less than capacity, levels of service are reasonable and the system can service the entire demand.
Region 2: Density values from vpmpl denote flow conditions in which traffic demands are approaching roadway capacity. Although the level of service has deteriorated, the system is actually more productive in terms of vehicle-miles of travel.
Region 3: Density values greater than 67 vpmpl denote flow conditions in which traffic demands exceed capacity. Not only are unacceptable levels of service encountered, but the system is not productive as to vehicle-miles of travel.

7. Determination of Density
Aerial Photography
Input-Output Counts
Speed/Flow Calculations
Percent Occupancy Calculations

8. Aerial Photography The earliest measurement technique
The speed of the airplane and length of roadway covered in each photograph determine the frequency of photographs.
Photograph overlap of percent is desired
Ramps, intersections and lane additions/drops are typically subsection demarcation points
The vehicles are manually counted on the photographs in each subsection.
Densities are then calculated and displayed on density contour maps.

9. Input-Output Counts
An initial count of vehicles along the roadway between two count stations are made.
Over time, the number of vehicles entering the section is continuously added, and the number of vehicles leaving the section is continuously subtracted from the initial count.
Section density is calculated on the basis of the difference between two large numbers (input and output counts). Frequent re-initialization is needed considering detector errors.
The input-output technique therefore is practical only if the detectors are very reliable, and accurate and an automatic initialization process can be employed frequently.

10. Speed/Flow Calculations
This calculation technique requires two detectors, one for counting and one for speed, or two closely-spaced detectors with software to convert elapsed travel time to speed.
One problem with this approach is that time-mean speed is calculated for the point measure of speed, and space-mean speed should theoretically be used in the calculation.

11. Calculation of Density from Traffic Stream Variables
q = flow (volume) [vehs/hour]
u = speed [miles/hour]
k = density (concentration) [vehs/mile]
Speed-Volume-Density Relationships (from dimensional analysis)
q = k ·us
us = space mean speed

12. Density vs. Lane Occupancy
An indicator of macroscopic density on freeway control systems is percent occupancy. Percent occupancy (or Lane Occupancy) is much easier to measure than density, and is defined as the percent of time a short section of roadway is occupied.
Density can be determined from percent occupancy

13. Percent Occupancy Calculations
In order to determine density from the measurement of occupancy, average vehicle length and detection zone length must be known.
In situations where vehicle lengths do not vary over time, the single presence detector for measuring occupancy is adequate.
In situations where average vehicle lengths are not known and vary over time, a second adjacent detector is required for calculating average vehicle length.
Single loop works only if vehicle length is known.

14. Total Time Detector Occ.
Lane Occupancy
Lane Occupancy is defined as the percentage of time a detector is occupied with a vehicle
Lane Occupancy = % (occ) =
(during time T)
= =100
1 minute
Time interval for aggregation = 30 seconds
20 seconds To T
Total Time Detector Occ.
Total Time Period
Real Time Control

15. Relationship Between Density & Lane Occupancy
Assume axle detectors
distance d
#1 #2

16. Speed of individual vehicle vi = =
Pulses #1 #2
Time
time ti
Rear axle
Front axle
Speed of individual vehicle
vi = =
distance time
d ti
ft sec

17. length of vehicle + length of sensor time of occupancy
B. Assume loop detectors LV LD to
Pulses
Time
length of vehicle + length of sensor time of occupancy
LV + LD to
ft sec
vi = =

18. Average speed will be us = in LV + LD to ft sec mph

19. C. If you look at a series of pulsestN
Pulse T
52.8
LV + LD
Then it can be shown that k = % Occ

20. D: Total time detector is occupied in time T
TO = t1 + t2 + … + tN in seconds
Average time each vehicle occupies detector
in seconds
in hours

21. us = us = N TO = 3600 LV + LD 5280 tO 3600 LV + LD 5280 3600 TO/N
N LV + LD us
us =
us = N
TO =

22. % occ = 100 % occ = knowing that us = and = q % occ = q = (LV + LD) kTO T
% occ = 100
% occ =
knowing that us = and = q
% occ = q
= (LV + LD) k
% occ = k
k = (% occ) N T
100 us
LV + LD 5280 q k N T
100
q/k
LV + LD
100
5280
LV + LD 52.8
LV + LD

23. Flow Conditions Based on Density and Percent Occupancy

24. Discrepancy Bet. Table 7.1 (HCM 85) & HCM2000 Values on Max. Density
LOS Table 7.1 HCM 2000 A B C D E
Reasons: vehicles getting shorter, vehicles following closer, more reliable vehicle technology, more congestion lead to more aggressive driving
No change from HCM 2000 to HCM 2010

25. Lane Occupancy and Flow

26. Introduce of Shock Wave

27. Density Contour Maps
Density contour maps are helpful in understanding traffic flow phenomena on congested uninterrupted traffic facilities.
Density contour maps also provide an introduction to shock waves and the data used in constructing these maps are essential in estimating travel times and traffic demands.
The first density contour maps were developed in 1960 in studies of Hollywood and Pasadena freeways.

29. Density Contour Map Fig 7.3
The horizontal scale is distance, with traffic moving from left to right.
The vertical scale is time increasing from bottom to top.
The array of numbers within the contour map are density values expressed as a number of vehicles per mile per lane in each subsection.
Uncongested or free-flow traffic conditions: densities <= 40 vpmpl.
Near-capacity flow conditions: 40<densities <= 60 vpmpl
Congested conditions: densities > 60 vpmpl.

30. Density Changes – Shock Waves
This one density of 40 and above are considered congested conditions.
Only the first digit is used.

31. Shock Wave
Definition: Flow-speed-density states change over space and time. When these changes of state occur, a boundary is established that separates the time-space domain of one flow state from another. This boundary is referred to as a shock wave.

32. Density Discontinuity=Shock Wave
Congestion zones are typically triangular-shaped with the boundaries of the triangle denoting significant changes in traffic flow conditions.
These boundaries are designated as shock waves since they represent discontinuities in flow conditions.

34. Single Bottleneck Shock Waves
(May Pg. 202, Fig.7.3)
Demand < Capacity 2 3
Time 1
Demand > Capacity
Traffic
Distance
1st Shock Wave – backward forming wave
2nd Shock Wave – forward recovery wave
3rd Shock Wave – stationary wave

35. System-Wide Bottleneck

36. Shock Waves at signalized Intersection

37. Shock Waves at a Bottleneck

38. Comprehensive Shock Wave Example

39. Comprehensive Example
At time 5:
First congestion at C
FS (5-6)
BF (5-9)
FS: Frontal Stationary
BF: Backward Forming

40. Comprehensive Example
At time 10:
Congestion due to bottleneck
FS (10-11)
BF (10-14)
BF (9-11)
Reduced velocity from (5-9)
Due to reduced flow to downstream from the bottleneck E
View 9-11 as the result of 5-9 overlapped with a forward recovering SW
FS: Frontal Stationary
BF: Backward Forming

41. Comprehensive Example
At time 6:
FF (6-1)
Due to slow moving trucks going upgrade
FS (1-2)
Truck speed stabilized
Truck speeds on grades
FS: Frontal Stationary
FF: Forward Forming

42. Comprehensive Example
At time 3:
Reduced demand from incident at B
FR (3-2)
Reduced demand from B to A
FS (3-4)
Bottleneck at incident site
FS: Frontal Stationary
FR: Forward Recorvery

43. Comprehensive Example
At time 4:
Incident at B cleared
BR (4-7)
BR: Backward Recovery

44. Comprehensive Example
At time 7:
Hit Bottleneck at C
FS (7-8)
FS: Frontal Stationary

45. Comprehensive Example
At time 12:
Bottleneck at E
FS (12-13)
FR (12-8)
Due to reduced flow from bottleneck at E
FS: Frontal Stationary
FR: Forward Recovery

46. Comprehensive Example
At time 8:
Congestion at C ends

47. Comprehensive Example
At time 14:
Demand reduced
Demand=flow
RS (14-15)
RS: Rear Stationary

48. Comprehensive Example
At time 15:
Demand further reduced
Capacity>demand
FR (15-13)

49. Comprehensive Example
At time 13:
Demand further reduced
Demand=Bottleneck capacity at E
Congestion at E ends
No congestion for the entire section

50. Comprehensive Shock Wave Example

51. Comprehensive Shock Wave Example

52. Shock Wave Classification

53. Shock Wave Classification
“Forward”
Moving towards downstream
“Backward”
Moving towards upstream
Direction
of Traffic

54. Shock Wave Classification
“Forming”
Queue gets longer over time
“Recovery”
Queue gets Shorter over time
Time

55. Frontal Stationary Shock Wave
Always be present at a bottleneck.
Indicates the location where traffic demand exceeds capacity.
It is due to recurring situations where each day the demands exceed normal capacities during the peak period at specific locations or may be due to nonrecurring situations where the normal demand exceeds reduced capacity (caused by an incident/accident) which may occur at any location at any time.
The term "frontal" implies that the boundary is at the front (downstream edge) of the congested region with lower densities downstream and higher densities upstream.
The term "stationary" means that the shock wave is not moving.

56. Backward Forming Shock Wave
Must always be present if congestion occurs (Very common).
Indicates the area in the time-space domain where excess demands are being stored.
The term "backward" means that over time the shock wave is moving backward or upstream in the opposite direction of traffic.
The term "forming" implies that over time, the congestion is gradually extending to sections farther upstream.
The time-space domain to the left of this shock wave has lower densities, and to the right, the density levels are higher.
The slopes of these shock waves represent velocities, with the flatter slopes representing lower velocities.

57. Forward Recovery Shock Wave
Also most commonly encountered.
Occurs when there has been congestion, but demands are decreasing below the bottleneck capacity and the length of congestion is being reduced.
The term "forward" means that over time the shock wave is moving forward or downstream in the same direction of traffic.
The term "recovery" implies that over time free-flow conditions are gradually occurring on sections farther downstream.
Densities are higher to the left of this shock wave, and lower to the right.

58. Rear Stationary Shock Wave
May be encountered when the arriving traffic demand is equal to the flow in the congested region for some period of time.
The term "rear" implies that it is at the rear (upstream edge) of the congested region with higher densities downstream and lower densities upstream.
The term "stationary" means that the shock wave does not change location over time.

59. Backward Recovery Shock Wave
Encountered when congestion has occurred, but then due to increased bottleneck capacity, the discharge rate exceeds the flow rate within the congested region.
The term "backward" means that over time the shock wave is moving backward (upstream) in the opposite direction of traffic.
The term "recovery" implies that over time free-flow conditions are extending farther upstream from the previous bottleneck location.
The congested region is to the left of the shock wave, and free-flow conditions are to the right.

60. Forward Forming Shock Wave
Not very common.
The term "forward" implies that the shock wave moves in the same direction as the traffic (downstream).
The term "forming" indicates that over time the congestion is extending to sections farther downstream.
To the left of the shock waves, densities are lower, while to the right, densities are higher.