Let us start with a familiar situation. Take a familiar part of a road, for example, the one from your home to work. How long will it take you to drive from one end of that stretch of the road to the other? At times you will have the road almost to yourself and will be able to drive at the maximum allowed speed without any problem. At other times you will end up in a queue that moves extremely slowly – in fact, you may be standing still for quite some time. Why?
Table of Contents
Creating a simple model
As in all physics problems, let us start by defining our problem set. We will start by assuming that our road is a simple lane (no room for passing). We also assume that our drivers follow the “1001 – 1002 – 1003” rule (meaning that there should be a 3-second interval between the cars on the road). We shall also only consider the steady-state situation, leaving the dynamics problem for a later insight.
Measuring the interval from the front bumper to front bumper
Figure 1. Two cars with a minimum interval
Figure 1 illustrates one way of measuring the time interval between cars. We have defined a “measuring point” on each car (here it is the front bumper) and measure the time interval between the front bumper of the first car passes an arbitrary point in the road to the following car passes the same point. Let us see what this measurement method implies.
- Since the minimum time interval between cars is 3 seconds, our measurement method implies that the maximum number of cars that can pass an arbitrary point on the road is one car every 3 seconds. One car every 3 seconds equals 20 cars every minute or 1200 cars every hour. The road capacity is therefore 1200 cars/hour
- Since the actual speed of the cars has not entered the calculations, we must conclude that the road capacity is independent of the speed limit. In other words: A local road with a speed limit of 50km/h has the same capacity as one lane on a highway with a speed limit of 100km/h.
- By the same reasoning, the maximum road capacity between two points depends only on the number of independent lanes, not on the speed limit. Thus, if one lane is blocked for some reason (accident, road work…), the road capacity is reduced accordingly.
Measuring from back bumper to front bumper
The measurement method given in Figure 1 has some serious problems at low speeds. One unintended consequence is that it assumes that all cars occupy the exact same space when they are standing still – which is of course impossible. In order to get better precision, we must modify our measurement method somewhat.
Figure 2. A better measurement method
Figure 2 shows a better way of measuring the time interval between cars. Now we will measure the time interval from the rear bumper of the first car passes an arbitrary point in the road to the front bumper of the following car passes the same point. This method solves the low-speed problem, but it is harder to analyze.
Figure 3. Road capacity as a function of queue speed
Figure 3 shows a calculation of the road capacity under the simplifying assumption that all cars are 5m long. From the curve we see that:
- From about 60km/h, the road capacity is largely independent of the average car speed.
- When average car speed falls below 30km/h, the road capacity falls drastically
Looking at the curve, we see that the road capacity at 25km/h is only 85% of the road capacity at 100km/h. One consequence is that a road that is at 90% of total capacity at 100km/h suddenly is overloaded if the average speed goes below 25km/h!
Some questions about road capacity
It may seem strange that the road capacity is more or less independent of the speed limit. After all, the time spent driving from A to B goes down as the speed limit goes up. So why does the capacity not increase?
The problem lies in getting onto the road in the first place. If the road is 100% full, you cannot get onto the road and start driving. If it is below 100%, you must still wait for an opportunity to get onto the road.
If the road is at full capacity, there are still no queues (by definition). So how does a queue form?
There are several reasons for queues forming. The most common is for cars trying to get onto a road that is already close to 100% full. Usually, this forces a car on the road to brake in order to keep the correct distance between cars. Since a speed reduction lowers the road capacity (Figure 3), this can suddenly cause the road to be more than 100% full.
Some observations about queues
Now assume that we have a queue on the road, a queue so bad that the cars are standing still. Assume that the average space a car occupies on the road when standing still is 5m.
Now take a collection of 50 cars that together utilize a stretch of road at full capacity.
- At an average speed of 80km/h, this collection occupies 3.6km of the road
- At an average speed of 50km/h, they occupy 2.3km
- And when they are standing still, they occupy 250m
Now, assume that the car collection drives happily along at 80km/h. Then something drastic happens, and the first car must go do a full stop. Going from 80km/h to a full stop (standard braking, no panic), takes about 14 seconds. The cars behind must also brake to a full stop, and since they come in 3s intervals, it takes about 160 seconds to bring the whole collection to a full stop. And, as noted above, they are now occupying 250m of the road.
Now assume that the first car is able to start up again, accelerating to full speed. When is the whole collection of cars back at 80km/h?
At this moment we need to consider the human element. The driver has to decide when it is “safe” to start up and increase the speed. This decision will vary among the individual drivers, but an informal measurement in front of the red light indicates that the average delay for such a decision to be made is about 1 second.
I have tried to show that the road capacity is almost independent of the speed limit and it depends almost exclusively on the number of lanes in each direction. What I have not touched is the dynamic response of a collection of cars on that road. So, who is going to do that?
Master’s in Mathematics, Norway. Interested in Network-based time synchronisation.