Optimize This: The Case for Mathematical Models of Public Transit

This post is the first in what I hope to be a many-part series called “Optimize This”, intended to be an introduction to mathematical modelling of transportation systems. I hope you enjoy this post enough to come back and check out more.

The goal of this post: To introduce you to the world of theoretical transit modelling, validate the field, and motivate you to listen to what models are able to tell us.

To appreciate this post: In order to follow some of the conclusions presented in this post, an understanding of fairly basic high school math is needed.

TL;DR: I make the case that mathematical models are a misunderstood but valid way of gaining valuable insights about transit with a relatively small amount of investment. I introduce the real-world concept of bus pairing, and go through a 1964 derivation that leads to an equation that describes this phenomenon. I conclude with 


Theoretical transit modelling exists mostly in the academic world. It is a thriving subject in the sense that new insights and papers are introduced with regularity. However, even the most fundamental results are largely ignored in the real world. While I’m sure there are many reasons for this, I have listed a few of the reasons I can think of below:

  1. Models are too idealized and simple – this is the very nature of a model, it fails to encompass every nuance of a certain scenario.
  2. Models dispose of common sense – often models will produce “optimal numbers”, which are often taken to be an inflexible “final answer”.
  3. Models are hard to understand – being a theoretical abstracted representation of the real world, models often rely heavily on mathematics and high-level logical concepts. This makes the subject difficult to approach for a person who is unfamiliar with the subject.
  4. With abstraction comes detachment – Often those who are responsible for making final decisions about public transit systems are in a political position or have deep-seated roots in the system they are trying to improve. This can lead to an emotional investment in certain areas, and models tend to reduce the complex to a simple equation, ignoring any of the well-intentioned hard work that may have been done previously. Reducing people’s emotion and physical investment to an equation can alienate they very people who are needed to make improvements a reality.

I will directly address all of the points above at the end of this article. First, I would like to introduce you to some basic public transit models.

Are Models Valid?

Before we start believing in the results produced by our models, it’s important to decide whether models are able to provide conclusions that are at all sensible in the first place. In order to do that, let’s see if we can find a model that can produce a result we know happens in the real world. In this case, I am going to discuss bus pairing, a phenomenon that I’m sure many people are familiar with. If a bus travelling along its route is delayed, it tends to become more delayed, mostly because more passengers will be waiting for it, and thus it has to handle more passengers. Conversely, the bus following this (now) late bus will run ahead of schedule, since it has to handle less passengers. These two effects cause the buses to converge and run in pairs. If you have ever waited for a late bus, only to find that two appear at the same time, then you have been a victim of this phenomenon.

For a mathematical example of this, I turn to one of the most fundamental papers on mathematical transit modelling, Maintaining a Bus Schedule written in 1964 by Newell and Potts1. In this paper, they present a very simplified version of a buses travel along a route. The basic equation describes the a bus’ departure time from some stop in relation to the same bus’s departure time from the previous stop, and the previous bus’ departure time at that stop, as well as the travel time between the two stops:

\displaystyle t_{mn} =(m+nk)\tau + nT

Okay. If your equation-reading skills are a bit rusty, it’s time to dust off the cobwebs and really understand this equation. Let’s go through the variables and terms one by one:

k is the ratio of the arrival rate of passengers over the boarding rate of passengers on the bus. This number is assumed to between 0 and 1, because if it was larger than 1 passengers would be arriving faster than the bus could load them, and the bus would never leave the stop. The understanding here is that the higher k is, the faster passengers are arriving.

τ is the time between each bus, called the headway.

m is the index of the current bus in question. If we are concerned with the 4th bus dispatched, we would replace m with 4.

n is the index of the current stop in question. If we are concerned with the second stop on the route, we would replace n with 2.

(m + nk)τ therefore accounts for the time between buses, and the time taken to load passengers. If buses run every 10 minutes, and passengers arrive 10 times slower than they board (k=0.1), then this term, for the 4th bus at the 2nd stop would be (4+2*0.1)(10) = 42 minutes.

T is the travel time between bus stops, assumed (in this paper) to be the same between all stops. Therefore, nT is how long it takes a bus to drive all the way to the nth stop. If there is a 5 minute travel time between all stops, then the driving time to the 2nd stop would be 10 minutes.

tmn is the departure time for a bus at the stop. Combining the two terms of our numerical example, we would conclude that the 4th bus dispatched will depart the 2nd stop on the route 47 minutes after the first bus departs the first stop on the route.

If you’ve made it this far, congratulations! I’m glad you’re still with me. You can see that despite the many simplifications this paper introduces, it takes some time to understand the concepts being combined. This is what makes mathematics so powerful in general: complex concepts can be expressed in terms of a relatively simple group of symbols.

Now that we have gone through that explanation, let’s skip over the rest of the derivation and jump to the result. At this point a variable α is introduced to represent a “bump” causing the bus to be late. The nature of that delay is not explained or elaborated on, just that it exists. As a result of this, a new departure time is derived:

\displaystyle \displaystyle t_{mn} =(m+nk)\tau + nT + \alpha \frac{(n+m-2)!}{(n-1)!(m-1)!}\left[\frac{k}{k-1}\right]^{m-1} \left[\frac{1}{1-k}\right]^{n-1}

The first part is just the same equation as above. Don’t be daunted by the second part, the important conclusion is in the first of the two square bracket terms at the end. Since we decided k was between 0 and 1, then k-1 will be less than 0, and the whole term inside the square brackets will be negative. However, as m increases, it alternates between even and odd. If you remember your exponents, any negative number to an even power will be positive, and this means the whole complex last term will alternate between positive and negative, and the departure time from a bus stop will alternatively increase and decrease as buses go by. In other words, buses will alternatively go behind and ahead of schedule, and you have the bus pairing described above.

This fundamental conclusion from a very simplified mathematical model reflects real-world behavior. Not only that, but the paper continues on to point out that k, the arrival rate of passengers at a stop relative to the boarding rate onto the bus is an important quantity. Even from one of the pioneering models, some important insights can be gained.

The Case for Models

Now it’s time to directly address some of the concerns I listed above:

  • Models are too idealized and simple – I would argue that the above model, one of the most basic ones in the field, is not simple. Idealized, yes – simplifying assumptions are made to make systems simpler and cleaner to express mathematically, but many other fields do this too, including physics. Part of the advantage of models is that they can be expressed and analyzed easily, and if a model was so comprehensive that it covered every nuance of a system, it would no longer gain the advantages of being a model: important results can be obtained with relatively little effort.
  • Models dispose of common sense – I’d like to rephrase that: models don’t dispose of common sense, they set aside certain aspects of common sense with the intention of applying it later on. Nobody is arguing that the results produced by a model is inflexible and absolute, rather that they can help guide the common sense in the right direction. If your mathematical model says bus stops on a route should be spaced 400 meters apart, you’re not going to place a bus stop in the middle of an intersection or an empty point on the route just because it’s 400 meters from the last bus stop. Common sense is invaluable; it can be aided by a model, not replaced.
  • Models are hard to understand – All higher-level topics have a certain barrier to entry, something that is common across all areas of understanding. If it was simple, we would already be using it and comfortable with it. Just because something is hard to understand is not grounds for its invalidation. True progress comes from hard work and a dedication to understand, even if it means working your way through the previous section.
  • With abstraction comes detachment – This is probably the hardest issue to address. In a sense it goes hand-in-hand with the second point, in that when you understand that models don’t dispose of common sense, you realized they also don’t dispose of hard work. If you’re in the position of having to make a large decision, one with many financial, social, and political implications, don’t you want to make sure you’ve used every tool in the toolbox? I certainly would.

Well, that’s about it. If you’ve made it through this entire article, I thank you, and commend you. I hope you have at least been a little convinced that mathematical models have some validity. Keep an eye out for more specific posts about transit modelling, as I hope to cover some topics such as dispatching and route design, as well as talk more about how we go about modelling something in the first place. If you’re a transit company reading this, I hope (if you don’t already) you will consider introducing or including mathematical models into your planning work. If you have nothing to do with transportation, I hope you are reading this with interest and will learn to incorporate some of the more abstract ideas presented here into your own life and work.

Happy moving!



  1. Newell, G. F., & Potts, R. B. (1964). Maintaining a bus schedule. In Proceedings of the Second Conference of the Australian Road Research Board (pp. 388–393).

3 comments on “Optimize This: The Case for Mathematical Models of Public Transit”

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