Microscopic Traffic Flow Theory

There are certain characteristics that are inherent to specific vehicles, as well as their drivers, which may have an influence on a traffic flow. In microscopic traffic flow models these individual characteristics are employed in order to describe the traffic flow in terms of the underlying interactions between drivers and their vehicles with one another [57]. Naturally, the behaviour of a vehicle in a given traffic environment is largely based on the behavioural aspects of its driver. For this reason, several models have been developed which are able to take varying driver behaviour into account in microscopic descriptions of traffic flow. The incorporation of human factors, however, greatly increases the model complexity [91] and, as a result, many microscopic traffic flow theories employ combined vehicle-driver combinations, modelling the vehicle and driver as a single entity in an attempt to reduce the model complexity.

Microscopic Traffic Flow Variables and Characteristics

When considering individual vehicles, several variables are typically associated with each vehicle in order to provide an accurate description of the traffic flow. These variables include the length of vehicle i, denoted by `i, the longitudinal position (typically taken to be the position of the

rear bumper [91]) of vehicle i, denoted by xi, the speed of the vehicle, given by

ui=

dxi

dt , and its acceleration

ai = dui dt = d2_x i dt2 .

Microscopic speed characteristics describe the speed properties of an individual vehicle passing a fixed point or a short segment of road during a specified time period [96]. Roadway design features, interrupted flow situations (e.g. stop streets, signalised intersections) and other road users make up the immediate environment which, in turn, affects the speed at which each individual vehicle travels. It is typically only the accelerating capabilities of the vehicles that directly alter their speeds, not other factors such as road and wind friction [91].

As in the macroscopic modelling paradigm, two other important characteristics of the traffic flow are the space and time headways. Time headway is often considered to be one of the most

important microscopic traffic flow characteristics due to its direct influence on the capacity of a road section [57]. The time headway hti of vehicle i is typically taken to be the difference in

the passage time of the rear bumper of vehicle i− 1 in front of it and its own bumper across a specific stationary point. The time headway therefore comprises a time gap tgi, which is defined

as the time taken for the front bumper of vehicle i to reach the current position of vehicle i_{− 1,} and an occupancy time toi, which is defined as the time required for vehicle i to traverse its own

length, that is hti = tgi+ toi [91]. Hoogendoorn and Knoop [57] referred to the time gap as the

net time headway, which is considered to be particularly important in respect of the analysis and modelling of the space requirements for overtaking manoeuvres. This type of analysis is also known as critical gap analysis, while the sum of the time gap and occupancy time is referred to as the gross headway.

space time vehicle i vehicle i_{− 1} xi−1 xi hsi xsi `i ti−1 ti hti tgi toi

Figure 3.2: A time-space diagram illustrating the trajectories of two vehicles (i− 1 and i), as well as their time and space headways. Adapted from Logghe [90].

Similarly, a space headway hsi is also associated with every vehicle i. The space headway is

defined as the distance from the rear bumper of vehicle i− 1, to the rear bumper of vehicle i [57]. As with the time headway, the space headway is also the sum of two components, namely the space gap xsi and the vehicle length `i. Again, the sum of these two components is typically

referred to as the gross space headway, while the space gap alone is known as the net space headway [57]. It is important to note that time headways are local, microscopic characteristics as they relate to the behaviour of individual vehicles, and are typically measured from a fixed point along a roadway, whereas space headways are instantaneous measurements taken at a given point in time. As may be seen from the definition of the expressions for the space and time headways, these two characteristics are highly correlated. This correlation is illustrated by the relationship hsi hti = xsi tgi = `i toi = ui. (3.7)

The relationship in (3.7) is illustrated graphically in a so-called time-space diagram in Figure 3.2. In the figure, the trajectories of two vehicles, i−1 and i, are traced out, showing their respective positions at every point in time. The speed of the vehicles is given by the gradient of the tangent to the line indicating each vehicle’s trajectory. In the case shown in Figure 3.2, the two vehicles

Car-following models have been established in the literature in order to capture the behaviour of one vehicle, following another along a specific road section, while incorporating the afore- mentioned microscopic traffic flow characteristics. Employing the notation presented above, a general car-following situation is depicted in Figure 3.3.

Direction of travel `i xsi(t) `i−1 xi(t) hsi(t) = xi−1(t)− xi(t) xi−1(t) ui−1(t) i− 1 ai(t + δt) ui(t) i

Figure 3.3: Car-following theory notations and definitions. Adapted from May [96].

In Figure 3.3, vehicle i is following vehicle i_{− 1 in a left-to-right direction. It is important to} note that the acceleration rate ai of the following vehicle is specified as occurring at time t + δt,

and not at t. The time duration δt represents a reaction time, required for the driver to react and subsequently apply the acceleration (or deceleration) rate [96]. The relative velocity of the lead vehicle and the following vehicle is denoted by ui−1(t)− ut(t). Given a situation where this

relative velocity is positive, the lead vehicle is travelling at a higher velocity, and as a result the magnitude of the distance headway between the vehicles is increasing. Conversely, if this relative velocity is negative, the following vehicle is travelling at a higher velocity, and the magnitude of the distance headway between the vehicles is decreasing. If the value of ai(t + δt) is positive,

vehicle i will start accelerating at time t + δt, with a negative value of ai(t + δt) indicating that

vehicle i will start decelerating at time t + δt. Finally, if the value of ai(t + δt) is equal to zero,

vehicle i is travelling at a constant velocity [96].

Various rules and theories have been proposed in the literature for governing when and at what rate vehicles should accelerate (or decelerate), based upon the above car-following model. Pipes’ [121] theory suggests that vehicles follow the guidelines set out in the California Motor Vehicle Code, stating that: “A good rule for following another vehicle at a safe distance is to allow yourself at least the length of a car between your vehicle and the vehicle ahead for every ten miles per hour of speed at which you are travelling.” The resulting expression for distance headway is therefore d_min= `i  ui(t) (1.47)(16.0934)  + `i, (3.8)

measured in metres. A comparison of the computed following distances with field data have shown that the computed values are sufficiently accurate for speeds ranging from 16–60 kilo- metres per hour, but significant differences were observed for speeds falling outside that range [96]. The approach adopted by Forbes and Simpson [37] considered the reaction time required

by the following vehicle to perceive the need for deceleration. As a result, the time gap between the lead and following vehicle should always be greater than, or at least equal to, this reaction time. Therefore, the minimum time headway should be equal to the reaction time added to the time it takes for the vehicle to traverse its own length. This relationship is given by

hti,min = δt +

`i

ui(t)

. (3.9)

Again, as with Pipes’ model, Forbes’ model performs well in the range 16–60 kilometres per hour. Forbes’ model outperforms Pipes’ model at speeds higher than 60 kilometres per hour, but it still shows significant errors when compared to the in-field test data [96]. A third example of car-following theory comprises the suite of models proposed by the General Motors research laboratory [26, 40, 41, 55]. These models are significantly more extensive and particularly important due to the wide range of accompanying, comprehensive field experiments, as well as the discovery of the mathematical bridge between macroscopic and microscopic traffic flow theories. All the models proposed took the general form

response = f (sensitivity, stimuli),

where the response was always represented by the acceleration (or deceleration) to be performed by the following vehicle, and the stimulus was always represented as the relative velocity between the lead and following vehicle. Varying representations of the sensitivity, ranging from a constant sensitivity to empirically calibrated sensitivity functions based on vehicle speeds, differentiate between the five models formulated at the General Motors laboratories [96].