Service Frequency Theory

Following on from the research that I have been doing for the past two years, I have come up with theory regarding public transport patronage and the impact of service frequencies thereon.

At present, it is still very much in the developmental stage: one day it may form the basis of a PhD. In the meantime, I post these notes here in the hope that someone may make some sensible comment.

The Theory

Essentially the theory is the non-linear nature of service level impact and falls into two parts.

Part 1 says that the function linking service frequency to patronage is not linear across a range of frequencies, but rather it varies according to the service frequency. If I were to take an educated guess, I would say that this distribution would look something like this:

You can see that at low levels of service frequency there is very low patronage and even quite large improvements in frequency generate little additional patronage. However, at some point the additional frequency generates quite a good deal additional patronage, until the frequency becomes very high, at which stage additional frequency makes little additional impact on patronage.

What is more, I think I am in the position to be able to put some actual numbers to this graph.

• Below 2 buses an hour there is little impact
• Between 2 and 3 buses an hour there is slightly greater impact
• Between 3 and 6 buses an hour there is a strong impact
• Between 6-10 buses an hour there is a slow down again
• Above 12 buses an hour there is little additional impact

These figures have been culled from a number of sources, which include

• Various consumer preference research which shows that people regularly state 'no public transport available' as a reason for not using public transport, even though there area is served - at a frequency of 1-2 per hour (fairly standard suburban bus services)
• My own research which shows that 4 services per hour (15 minute headway) has a statistically signifcant (negative) correlation with private car use
• Research (of some age) that shows that arrivals at stops become random at around 6 services per hour (implying some sort of behavioural change at around this level)
• Intuition

When I had a look at this curve, I found that it fitted rather well the curve generated by the function

y=exp(4x)/(1+exp(4x))

This is of course a standard logistic function, and thus suggests possibilities for futher investigation.

The second part of the theory is somewhat more unusual. It states generally speaking that the mode share of public transport at any given time of day is determined not only by the service frequency at that time, but also the service frequency provided at other times. For example, it suggests that the choice of mode for journey to work is affected not only by the peak hour service frequency, but also by the availability of public transport in the evening.

The hint that led to this theory is provided in the following chart:

It shows that the areas served by Sydney Buses by and large have lower use of cars than those served by private buses. One of the key differences between the two is that the public buses offer services at night and more frequently on Sundays, while private buses rarely operate at night, and infrequently (if at all) on Sundays.

There is little in the literature either way on this proposal, but there is some anecdotal support. For example at least one manager of a major bus company has said that in order to support daytime patronage they must provide night and weekend services, even if the latter are not particularly well used.

My own research also shows that the previously mentioned 4-per-hour frequency of service shows significant negative correlation with vehicle use only when supported with 2-per-hour service frequency at night and on Sundays.

Obviously, there is a good deal of research to be done here.

Implications

The implications of this theory are clearly quite strong.

Currently, where systematic transport planning is undertaken there is probably some model used to predict various mode shares under various scenarios. While some of these models are statistically quite sophisticated, under the hood they are all essentially attempts to project into the future using current data as a starting point. Most will have some sort of discrete choice model which have at their heart a utility function that looks like

Ux = f(A) + f(B) + ... + e

Where Ux is the utility of chosing mode x, f(A), f(B) etc are the deterministic functions representing the various attributes of the trip, passenger and mode, and e the error term (about which we may make many assumptions, particularly regarding distribution).

If B were service frequency we would probably find that within the model it would simply expand to the linear form

f(B) = aB+k

If the relationship is non-linear (as part 1 of the theory suggests) then this function will generate erroneous results. Similarly, using a simple frequency, rather than an compostite frequency (as part 2 of the theory suggests) then results will be inaccurate.

What is more given that the the data in the model is based on current service frequencies (1-2 per hour) then they may be significantly under-estimating public transport mode share potential.

In the short term, I am continuing to recommend the "15/30 standard" (4 per hour daytime, 2 per hour night and Sunday) as a standard basic service frequency for public transport given that this has been shown to be efficacious, and in the longer time, more research is clearly needed.