Flow characteristics

The interruption to traffic flow at a signalized intersection is orderly and deterministic. Consider the following scenario. The signal has just turned red for a particular stream or movement. All vehicles on this stream come to a stop and remain stopped (thereby forming a queue) till the light turns green. Once the light turns green the first vehicle in the queue departs followed by the other vehicles. Movement of vehicles continue unabated till the light turns amber at which point vehicles close to the intersection generally go through while the ones farther away from the intersection initiate maneuvers to come to a stop. The same pattern follows for every cycle. Given this interruption pattern, the following processes become important for analysis: (i) the arrival process of vehicles (ii) the departure process of vehicles (iii) the queue of vehicles, and (iv) the delay to vehicles.

In this section, the first two processes are described. The next section analyzes the other two processes. The last section in this topic analyzes the related matter of capacity and level of service at signalized intersections.

Arrival process

Arrival processes at intersections could be of three kinds (i) random arrivals, (ii) grouped arrivals, or (iii) mixed arrivals. In random arrivals vehicles seem to arrive at the intersection randomly. Such arrival pattern is seen at isolated intersections, i.e., intersections at locations where there are no other upstream intersections in the vicinity (say within 3 to 4 km). In these cases the inter-arrival times (or headways) are often distributed more or less according to the negative exponential distribution. That is, if $h$ is headway between vehicles, then $P(H_1 \leq h \leq H_2)$, the probability that a particular headway is between $H_1$ and $H_2$ is given by

$$P(H_1 \leq h \leq H_2) = e^{-\lambda H_1} - e^{-\lambda H_2}$$ (2)

where, $\lambda$ is the reciprocal of the mean or average headway. The above expression is derived by integrating the negative exponential probability density function (for headways in this case) between $H_1$ and $H_2$. The negative exponential probability density function, $f(h)$ (for all $h>0$) is given as

$$f_h = \lambda e^{-\lambda h}$$ (3)

An assumption (or observation) of negative exponential distribution for headways also implies that the vehicle arrival process is a Poisson process. That is, $P(N_t=k)$, the probability that the number of vehicles that arrive in a time interval $t$, $N_t$, is equal to $k$ is given by:

$$P(N_t=k) = \frac{(\lambda t)^k e^{-\lambda t}}{k!}$$ (4)

where, $\lambda$ is the reciprocal of the mean or average headway or, alternatively, it is the mean or average arrival rate.

Grouped arrivals are seen at intersections which are located close to (say within 2 km) another upstream intersection. In such cases, arrival process seems to be uniform and vehicles can be assumed to arrive at reasonably constant headways. This phenomenon occurs because vehicles arriving at the intersection are the ones which have been released by an earlier intersection and therefore are in a platoon.

Mixed arrivals are seen at intersections which are located at intermediate distances (say between 2 to 4 km) from another upstream intersection. Here the arrival cannot be characterized either as purely random or as purely grouped. This is because the distance, between the upstream intersection and the intersection being studied, is large enough for many of the released vehicles to disperse from the discharged platoon and arrive independently; yet the distance is not large enough for the entire platoon to disperse, hence some vehicles still arrive in a grouped manner.

Departure process

When a signal turns green from red the first of the stopped vehicles initiate maneuvers to move and cross the intersection. Next to cross the intersection is the second vehicle in the queue and so on. If one measures the headways, i.e., the time gaps between successive vehicles when they cross a pre-specified point on the intersection (generally the stop line on the road), then an interesting and expected pattern emerges. Figure 6 shows a typical plot of headways versus position in queue. The ordinate value for an abscissa value of $i$ gives the headway between the $i^{th}$ and $(i-1)^{th}$ vehicles in the queue when they cross the pre-specified point in the intersection. Further, the ordinate value for an abscissa value of one indicates the time-gap between the light turning green and the first vehicle crossing the pre-specified point in the intersection.

図 6: A typical plot of departure headways versus position in queue at a signalized intersection.

From the figure, two features emerge, (i) the headway stabilizes to a value, $h_s$, referred to as the saturation headway; this value basically states the maximum number of vehicles that can ever be released during a specified green time, and (ii) the initial headways are larger than $h_s$. The second feature highlights that although vehicles can move at a headway of $h_s$, the initial vehicles take a longer time due to perception / reaction time (to the light turning green) and the extra time taken to accelerate to a reasonable speed (note that the latter vehicles more or less achieve this speed when the cross the specified point as they start moving from a distance further upstream from the specified point. In a sense then, some time is lost due to the fact that the initial vehicles take longer than $h_s$. The sum of these excess times is referred to as the start-up lost time, $l_s$. That is,

$$l_s = \sum_{\forall i} \left( h_i-h_s \right)$$ (5)

Near the end of the departure process some time is also lost. This happens because invariably some part of the amber time remain unutilized because vehicles come to a stop even when some part of the amber is still remaining. This loss of time, referred to as movement lost time (or sometimes as clearance lost time), $l_m$, is primarily due to the fact that drivers are never aware of the remaining amber time.