The Problem: Bus-bunching

It is hard for buses to adhere to a schedule because there is a tendency to bunch. The result is that riders experience a long wait, after which several buses may arrive together.

Worse, a schedule is inflexible. If a bus breaks down it leaves a gap in service and this gap tends to grow. The bus following this gap will encounter more passengers and so will spend more time at each stop, and fall still farther behind.

Bunching is the most frequent complaint about any urban bus system, as a web search on “bus bunching” will confirm.

A Solution

Abandon the schedule! No one cares about a schedule as long as the gaps between buses — that is, the headways — are small, as in a busy urban bus system. Instead control headways by strategically delaying buses at the ends of the route or at special locations such as transfer points.

John Bartholdi (Georgia Tech) and Don Eisenstein (UChicago) have devised a new way of computing the delays so that headways equalize, without management direction or driver intention.

How it works

Our scheme requires only this: When a bus arrives at one end of the route, it “looks” at the bus immediately following and estimates the time until its arrival. Then it performs a simple calculation that determines how long to pause. This pause changes the headway of each newly arrived bus to an average of its former headway and the headway of the following bus. If its former headway was larger, its new headway becomes smaller, and vice versa. The result is that headways are constantly adjusted to become more nearly equal — and without anyone having to guess in advance the eventual common value.

The scheme requires only a tidbit of local information — the estimated time until the next bus arrives — but this is sufficient to coördinate all the buses on the route. The relative positions of the buses will be adjusted to be more evenly spaced, so that no one has to wait too long for a ride. Moreover, this technique works without knowing a map of the route or even the number of buses.

Read the details in our technical paper. The formal version, which appeared in Transportation Research B, won the 2012 “Best Paper Award” given by the Transportation Science and Logistics section of the professional society INFORMS.

Live Simulation of Buses on a Route

Below is a live simulation of buses on a route. The simulated buses will likely have achieved equal headways by the time you have read this far. Click anywhere on the simulation to remove one bus, leaving a large gap, and you will see that the remaining buses spontaneously re-equilibrate. In general our system can transition from n buses to n-1 or n+1 without any need for management oversight or coördination among the drivers.

Click anywhere on the simulation to remove one bus


Without changing operations or processes, buses can be added or removed from the route at any time; the route can be changed (for example, to detour around construction); and bus stops can be inserted or removed. After any such change the headways will autonomically re-equalize.

Bus route headways were both shorter and less variable under our scheme

Figure 1: In tests on the Georgia Tech bus route headways were both shorter and less variable under our scheme (red) than on similar days in which buses tried to follow a schedule (gray).

Success on a real bus route

Thanks to the bravery of David Williamson and Aaron Fowler of Georgia Tech’s Department of Parking and Transportation, we were able to test our scheme on the central bus route through the heart of campus. The results were clear (Figure 1): Average headways were smaller so there was less wait for a bus. And there was less variability in headways, so service was more reliable.

More importantly, tests confirmed the ability of our scheme to respond to large disturbances. Figure 2 shows what happened after we removed one bus from the route, leaving a large gap in service. Under our scheme, the headways of the remaining buses spontaneously re-equalized, thereby re-establishing regular service without intervention by management or awareness of the drivers.


Our scheme is simple and practical. It is easy to try, easy to implement, and easy to adjust to special circumstances. We also expect it to be useful for other types of transportation with short headways, such as subway trains or airport shuttles.

Frequently Asked Questions About the Idea

What is special about this method of headway control?
Its simplicity and practicality. Public transit is not an experimental science and managers of transit systems are understandably reluctant to risk failure — but our scheme is easy to understand and it is cheap, easy, and safe to try out. And it works.
Don’t buses already have GPS installed? How is this different?
Yes, most buses have Global Positioning Systems (GPS) installed. But our scheme does much more than simply track locations of the buses: Our contribution is a mechanism that automatically adjusts the positions of the buses so that they are equally spaced in time; that is, they arrive with the same headway (time between arrivals). This provides the best service and gets maximum use from each bus.

Evolution of headways after removal of a bus from the route

Figure 2: Evolution of headways after removal of a bus from the route: Under our scheme the buses reestablished equal headways spontaneously, without intention or even awareness of the drivers and without intervention by management.

Isn’t dynamic headway control an old idea?
Others have suggested various schemes to adjust the headways, but ours abandons not only the idea of a target schedule but also the idea of a target headway. A preconceived headway is equivalent to a preconceived bus velocity — but there is little control over bus velocity in the real world because, except in special cases, it depends on surrounding traffic and intensity of ridership. Because we focus on the essential, our scheme is extremely simple and practical. Schemes that chase target headways attempt to control the uncontrollable. They require much more data, such as instantaneous bus locations and velocities, rates of passenger arrivals to each stop, queue lengths, and so on, and still miss the mark.
How is this different from “time frequency scheduling”, such as is used on the DC Circulator?
Time frequency scheduling is the setting of a target headway (rather than a target schedule). It is a step in the right direction but still suffers from the problem that, except in unusual circumstances, the system manager cannot control traffic velocities. It is fine to have a target headway, but the question is whether it can be achieved. It is not enough to show the drivers where the other buses are, and expect them figure out how to fix problems. Our system abandons target schedules or headways as distractions, and focuses on the essential: that headways be as nearly equal as possible. Furthermore, no one has to figure out how to fix an imbalance because headways equilibrate spontaneously. (Incidentally, the DC Circulator uses a schedule; the schedule, however, is not published.)
Can’t the same thing be achieved if the drivers can communicate, say by walkie-talkie?
No. Each individual has only a local view of the route and may, in trying to make local improvements, make overall performance worse.
Is this scheme suitable for routes with long average headways?
Probably not: Research has shown that people want a schedule if headways are longer than 10–12 minutes.
Does this scheme coördinate the buses with those of other routes to which passengers may want to transfer?
No: Our scheme coördinates the buses on a route but not among routes. But if headways are short, this should not matter.
Won’t passengers be annoyed if the bus waits at a control point?
We recommend choosing endpoints of the route as control points because there will generally be no on-board passengers. Otherwise, a good choice of control point is any bus stop where passengers can change to other transportation, such as other routes or trains. At such stops passengers generally appreciate the waiting time, which increases their opportunities to make seamless connections.
Where should control points be located?
The ends of an out-and-back route are natural locations for control points, because buses can pause there without delaying in-transit passengers. Other natural locations are wherever the route intersects with other transport modes, such as train or other bus routes.
How many control points should there be?
There is a tradeoff: More control points provide more assertive control; but buses pause at each control point and so more control points mean more idle bus capacity. There is no “right” answer but rather a management decision. Fortunately, it is easy to experiment.
Your scheme requires a forecast of time until the next bus, but this will never be accurate. What is the effect of inaccuracy on the performance of your scheme?
The mathematics shows that our scheme will continue to resist bunching as long as the forecasts are not frequently and wildly off-mark.
How does bus capacity affect this scheme?
When buses are bunched, the trailing buses can be nearly empty, which reduces effective capacity of the system. Because our system keeps buses well-spaced, it tends to spread the load among buses and so make best use of capacity.
You have proved a theorem that shows convergence to a common value of headway, but this theorem describes an idealized set of buses. What does it mean for the real world?
Because of inherent and ineradicable variability in traffic velocities, we do not expect real buses to achieve exactly equal headways. Instead, we interpret the theorem to mean that our scheme resists bunching. We expect that bus headways will vary less under our scheme than under a schedule and that gaps in service will not tend to grow, as they do under a schedule. These expectations have been confirmed in simulations and in daily operation on real bus routes.