1. Microscopic Density Characteristics
Microscopic density are represented by the longitudinal spacing characteristics
Chapter focus is spacing
Particularly important to safety, capacity, and LOS
Minimum spacing for safety
Spacing v.s. capacity (do not forget speed)
Larger spacing better LOS
I do not like the term of micro spacing.

2. Spacing=Distance Headway
The longitudinal space occupied by individual vehicles consists of:
(1) the physical space occupied by the vehicle (vehicle length), and
(2) the distance gap (clearance) between the vehicle and either the vehicle immediately ahead or immediately behind the subject vehicle.
dn+1(t) = distance headway of vehicle n+1 at time t (feet),
Ln = physical length of vehicle n (feet), and
gn+1(t) = distance between vehicle n and vehicle n+1 at time t (feet)
Note: Spacing and clearance are more commonly used in newer literature

3. Distance Headway vs. Time Headwayx h = d n 1 +
hn+1 = time headway of vehicle n+1 at point p (seconds), and
vn = speed of vehicle n during the time period hn+1 (fps)
The time headway is easier to measure by a roadside observer, and the distance headway can then be calculated
Same is true for detector measurement

4. Density vs. Spacing
Density is defined as the number of vehicles occupying a single lane of a length of roadway over a distance of one mile
Unit is vehicles per mile (per lane)
dn: individual distance headway (spacing)
N: number of observed distance headways

5. Spacing and Car-following
Car-following leads to spacing characteristics
Car-following considerations:
Spacing in front
Safety
Desired speed
Separation w.r.t. speed and flow
Flow also a factor

6. Car-Following Follow-the-leader model started in 1950s
Currently 3 categories of models:
safe-distance models
stimulus-response models
psycho-spacing models

7. Car-Following Notations

8. Car-Following Model Basics
Acceleration/deceleration of the trailing vehicle is specified at time t + Δt (not t). The Δt term is the interval of time required for the driver of the trailing vehicle to decide that acceleration/deceleration is necessary. This element of time can be termed as a reaction time.
The distance headway is simply the distance between the lead and trailing vehicles at the same point in time,
The relative velocity of the lead and trailing vehicles is simply the difference of vehicle velocities at the same point in time. A positive relative velocity indicates that the lead vehicle is pulling away from the trailing vehicle, and a negative relative velocity indicates that the trailing vehicle is gaining on the lead vehicle, and
Acceleration can be positive or negative (deceleration).

9. Safe-Distance Car-Following Models
Safe-distance car-following models describe the dynamics of a single vehicle in relation to its predecessor.
Pipes’ Model
Leutzbach Model
Forbes’ theory

10. Pipes’ Model
Pipes’ Model (1953): “A good rule for following another vehicle at a safe distance is to allow yourself at least the length of a car between you and the vehicle ahead for every ten miles an hour (16.1km/hr) of speed at which you are traveling”
1.47=5280/3600
With a different L, the number 1.36 will change
X is in mph
Recall your lab. Any objection to Pipes’ model? Also your experience.
For a vehicle length of 20ft:

11. Pipes’ Model Min. spacing changes with speed
If headway-based, for normal regime, a maintain a constant headway is a possible CF model.
Min. spacing changes with speed
Min headway does not change as much
Is hmin too large?

12. Leutzbach Model
Leutzbach (1988) discusses a more refined model describing the spacing of constrained vehicles in the traffic flow.
He states that the overall reaction time T consists of:
perception time (time needed by the driver to recognize that there is an obstacle);
decision time (time needed to make decision to decelerate), and;
braking time (needed to apply the brakes).
Assume stonewall effect of the leader
T: reaction time

13. Forbes’ Theory
Consider two successive vehicles: if the first vehicle stops, the second vehicle only needs the distance it covers during the overall reaction time T with unreduced speed, yielding Forbes’ model.
For a vehicle length of 20ft and a reaction time of 1.5 sec:
Leutzbach: Spacing, also includes deceleration distance.
Forbes’ headway,
Forbes: considering reaction time and vehicle length, the follower would be safe (dose the same thing as the leader). May not be true if the leader “hit the wall” turns to be true for normall driving but not for emergency situation.
Dt: reaction time

14. Forbes’ Theory Min. spacing changes with speed
Min headway is a constant unless speed is very low
Two regime.
Min. spacing changes with speed
Min headway does not change as much
Is hmin too large?

15. Smaller hmin?

16. Stimulus-Response Car-Following Models
In general, the response is the braking, deceleration, or the acceleration of the following vehicle, delayed by an overall reaction time T.
Stimuli
Spacing
Speed Difference
etc
Sensitivity are parameters to be calibrated.

17. Chandler (1958), Gazis (1961) Chandler et al :
where vn (t) and an(t+T) respectively denote velocity and acceleration of vehicle n at t and t+T, and g denotes the driver’s sensitivity. Thus, the stimulus is defined by the velocity difference between leader and follower.
Gazis et al.:
Thus, the following vehicle adjusts its velocity vn(t) proportionally to both speed and distances differences with delay T. The extent to which this occurs depends on the values of c, l and m.

18. GM Models- 1st Generation
1st model only considers relative speed

19. GM Models- 1st& 2nd Generations
2nd model proposed 2 states for sensitivity factor, higher value for low spacing (Fig 6.5)
Reaction time and sensitivity do not directly correlated?

20. GM Models- 3rd Generation
3rd model considers spacing

21. GM Models- 4th Generation
Improving the sensitivity by introducing the speed of the following vehicle.

22. GM Models- 5th Generation
5th model considers non-linear responses

23. Psycho-Spacing (Psycho-Physical) models
The basic behavioral rules of such so-called psycho-spacing models are: At large spacings, the following driver is not influenced by velocity differences.
At small spacings, some combinations of relative velocities and distance headways do not yield a response of the following driver, because the relative motion is too small.

24. Psycho-Physical Models
Models based on driver’s perception
Driver switch from one model to another when a certain threshold is reached
Wiedemann (1974) developed a Psycho-Physical model, that was used in VISSIM later

25. Other Car-Following Models
Optimal Velocity Model
Follower’s acc/dec depends on his/her speed and the optimal speed he/she can attain
Fuzzy logic models
Based on human perception of environment, rules such as “if distance divergence is too far and relative speed is closing then no action”
Neural Network Models

26. Traffic Stability
The stability of the traffic stream can be defined by the reaction time being used.
High reaction times (sluggish behavior) generally result in exaggerated responses: acceleration/deceleration rates. This would define regions of "unstable" behavior, characterized by high values of reaction time and sensitivity response.
"Stable" behavior is further characterized by moderate values of both reaction time and sensitivity response.
If the product (C) of these two parameters (Δt and α) is high, unstable conditions are likely to occur, while if the product is low, stable traffic conditions will occur.
Related Paper: Influence of Reaction Times and Anticipation on the Stability of Vehicular Traffic Flow, Treiber, Martin ; Kesting, Arne ; Helbing, Dirk, Transportation Research Board 86th Annual Meeting, 2007

27. Traffic Stability
The GM research defined two regions of traffic stability:
Local stability: involved with the car-following behavior of just two vehicles.
Asymptotic stability: concerned with the car-following behavior of a line of vehicles.
In order to define boundary conditions for these two levels of stability, the product (C) of reaction time (Δt) and sensitivity (α) was calculated and tested in the field.
The resulting values of C are provided in Table 6-4 (page 180).

28. Traffic Stability

29. Stability Illustrated

30. Car-Following Data
“Wired vehicles” in 1950

31. Car-Following Data
Single Loop Detector

32. Car-Following Data
Double Loop Detector

33. Data from Controlled Experiments (Example)
Single Lane, 1080m
No Overtaking
No Considering of Curvature
85 vehicle gradually Enter/ Exit every 20 s
Detectors at 270m, 540m, 810m
Increased demand stage (1700s)
Constant demand stage (200s)
Decreased demand stage (1700s)
Source: J.Wang, University of Leeds

34. Model Development Data, data, data Calibration of parameters
Minimizing errors based on a selected variable (micro/macro)
MLE,GA, etc.
Calibration vs. validation

35. Other Car-following Studies
Other data collection devices, such as GPSs
Other models, such as NNs, Fuzzy Logic,
Car-Following Model Based on Artificial Neural Networks in Urban Expressway Sections, Liu, Xiaoming; Wang, Li; Zhong, Xiaoming , Transportation Research Board 85th Annual Meeting, 2006
Study on Chaos Model of Expected Space Headway in Urban Expressway, Wang, Li; Yang, Xiaokuan; Zhang, Zhiyong; Du, Zhencai , Transportation Research Board 85th Annual Meeting, 2006
P. Chakroborty and S. Kikuchi. Calibrating the membership functions of the fuzzy inference system: instantiated by car-following data. Transportation Research Part C, 11:91–119, 2003.
C. Kikuchi and P. Chakroborty. Car following model based on a fuzzy inference system.
Transportation Research Record, 1365:82–91, 1992.

36. Lane Change Lane change also affects headway
Car following and lane change are core logic of microscopic simulation
Car-following model may consider lane change; lane change should consider car-following
“ghost vehicle” during lane changing process in simulation
Ghost vehicle?

37. Lane-changing?
“Hiatus” period

38. New Logic for Autonomous Vehicles
ACC: adaptive cruise control
automatically adjusts the vehicle speed to maintain a safe distance from vehicles ahead
On-board sensors (Radar or Lidar)
CACC: corporative adaptive cruise control
 the preceding vehicle's acceleration is used
obtained from the Cooperative Awareness Messages it transmits using DSRC or WAVE 
Lane changes
Truck platooning

39. Other New Developments
Speed, position data from Connected Vehicle (CV) technologies
New data, new model, new parameters (i.e. sensitivity, reaction time, etc.)
Calibration and validation?