There is a lot to be said about timetables. Buses in particular have a complex relationship with their timetables that can differ between countries, cities and even operators within the same city. In the UK, where I live, there are usually timetables provided for passengers to help them plan their journeys. In rural and suburban areas, a bus may come at a very specific time, while urban and other busy areas may have timetables specifying that buses on a route should arrive at some constant interval. On the other hand, when I was travelling in Rwanda in 2011, I was impressed by the fact that many of the buses did not run according to any timetable at all! Buses left when they were full rather than at some specified time. These differing strategies each have their benefits and disadvantages. Having a clear timetable with buses arriving on time can help an individual plan their journey well and use their time efficiently. Filling up buses before sending them on a journey may not be as efficient for individuals who want to use their time well, but it is a strategy that can be efficient both economically and environmentally. With a full bus there can be both a greater profit for the operators and lower fares for passengers, while the passengers' carbon footprint is reduced.

While I thoroughly enjoyed travelling in Rwanda, I am not going to spend my time trying to make authorities abandon timetables (at least not presently). Instead, I am going to discuss a particular type of timetable - the kind where passengers are told that a bus should arrive every x minutes. The thing that has really piqued my interest about this particular type of timetable was this article about the waiting time paradox. The waiting time paradox is a particular case of the inspection paradox. This paradox occurs when the probability of observing a quantity is related to the quantity being observed. In the special case of the waiting time paradox, this means that when a person arrives at a bus stop, they are more likely to be waiting for a late bus than for an early one. This is because there will be a higher chance of arriving during the longer time span between buses in the case where a bus is delayed than in the shorter time span when the next bus will be early. If buses on average arrive every x minutes at a bus stop, the average time that one will have to wait for a bus will therefore likely be more than the x/2 one could naively expect.

In fact, we can consider a simple scenario. Let us consider a bus stop at which a bus should arrive every 10 minutes. One could therefore think that the average time one would have to wait would therefore be 5 minutes. In fact there are two buses that are soon going to arrive, one in 15 minutes followed by a bus that arrives 5 minutes after that. The average bus arrival time is therefore (15 + 5) / 2 = 10, so the buses have not strayed too far from the schedule. However, if a person arrives at random during these 20 minutes, the chance that they will be waiting for the delayed bus is 15/20 and the chance that they will be waiting for the early bus is 5/20. Therefore, the time one could expect to wait at this bus stop would be:

15/2 x 15/20 + 5/2 x 5/20 = 6.25mins

So we can see that delays on a bus service translate into an increase in the average waiting time, even if there are buses that speed up and arrive earlier to try and counter this fact, keeping the average arrival time steady. In fact, it can be shown that the average waiting time should increase proportionally to the square of the average delay of buses on the route.

Term Meaning Average arrival time The average time between buses arriving at a station Average waiting time The average time you would have to wait for a bus if you arrived at a bus stop

Transport for London's schedule keeping

Out of interest, I have tried to investigate what effect the waiting time paradox has on Transport for London's buses. Below is a live chart showing bus data as it is happening right now. The number on the left is the bus route. The blue line represents how often a bus should arrive at bus stops at the current time (TfL usually specifies a time range - e.g. 9 to 13 minutes). The red line represents how much time one would expect to have to wait for a bus (i.e. half the arrival times of buses). The green data represents current data as has been acquired from the TfL open API. The green rectangle with the line through it represents the lower quartile, median and upper quartile respectively. The filled green square is the mean waiting time at bus stops for the next bus on that route at the current time. The maximum current waiting time on the route is represented by the black filled square and attached text.

Please bear in mind that this data is sampled every 10 minutes to reduce pressure on the TfL open API. The data is also not complete - it includes only Bus Routes 1-50 and only displays those routes where the timetable states that a bus should arrive every x minutes at the current time. The TfL open API uses GPS data to track arrival times, so they are fairly accurate. However, there is always a possibility for error; I was once told that a bus would arrive in 3 minutes, but I never saw it arrive at all - no technology is perfect. Overall, this API is still the most accurate data I can access without leaving my chair.

Powered by TfL open data

At the time of writing (5th May 2019), Transport for London are doing rather well when this data is analysed. A vast majority of passengers waiting at bus stops will see their buses arrive within or under the expected waiting time. In fact, the average bus on the 5th May 2019 has arrived in only 92% of the expected waiting time.

What does this mean when we consider the waiting time paradox? Why are the current waiting times not (necessarily) longer than the expected waiting times? What we are looking at is a timetable that does not tell us an objective truth.

Timetables can serve multiple purposes, not necessarily just a statement of how often buses factually arrive. We can prove mathematically that the waiting paradox is true. If most real waiting times are lower or within the expected waiting times, what is clearly happening is that the timetable is stating that buses arrive less frequently than they actually do. There are many reasons to do this - the main one being to ensure that passengers' expectations are met. If buses are more frequent than the timetable states, then the quotas stated in the timetable are more likely to be met.

In this case, the timetable is not in any way a source of truth. It is some form of contract that specifies what criteria the bus operators should meet to keep the passengers happy. This is a bit of a bizarre observation, especially since most people will feel that they have never been asked what would make them happy. Few people will have had an input as to how frequent buses should be on a route (other than by voting via the ticketing system a.k.a supply and demand), yet here stands some form of contract that bus operators will try to meet by adding buses to the route or by regulating the service.

As wonderful as this may be for those of you who often wait for buses, what happened to truth? Should timetables reflect reality, an ideal we are striving towards, or a minimum that we should always exceed? With the availability of live bus data on screens at bus stops and on websites, these questions may be slowly becoming irrelevant, but these questions will always remain in other walks of life where we use metrics to measure performance or expectancy. Should such questions be treated differently in transport than elsewhere? I cannot say I know the answers, but the debate is certainly interesting.

published: Fri May 03 2019