Crowds, delays and late-running - Part 2

It's only slightly harder to find a parking spot in a car park that's 91% full than one that's 90% full. It might take 33 seconds instead of 30 seconds, for example. However it's twice as hard to find a spot where it's 99% full instead of 98% full. Finding one would take twice as long on average. And when it gets really full, from 99 to 99.9%, you'd expect parking would take 10 times longer, since there's only 1/10 the number of free spaces.

This is the rule that even a tiny increase in usage of a system nearing capacity causes service faults (or more accurately, their impact) to rise exponentially.

In Melbourne we are currently experiencing the same on our peak-hour trains. Operators struggle to meet performance targets and rising patronage is cited as the key reason why. Today's stats compare to five years ago, where the likelihood of your train being late or cancelled was 50 to 70% less, but patronage was also lower, and if your train was cancelled you were more likely to be able to board the next one.

Studying lateness and cancellation figures is useful but don't always tell the full story, especially as seen by the passenger. In some cases a cancelled train may be effectively replaced by the previous running late, and the overall impact might be negligible. But in others, even a 2 minute delay could cause buses or other trains to be missed, resulting in 15 - 60 minute delays. Similarly the cancellation of a busy service might lengthen waits as passengers are unable to board the next three or four crush-loaded trains.

For simplicity, let's assume that trains can carry 1000 people crush-loaded. An on-time morning peak train approaching Station A already has 800 on board. As 80 people board at this station, all can board.

The same is also true for the next station (Station B). By the one after that the train will be full, but assuming some alight (eg a nearby school is a popular destination) then Station C's 80 people should also just be able to fit on.

The situation changes if the train departs late.

For a start there will be more on the platform at previous stations, as some intending to catch the following train will have started walking on. The total number of people walking through the validators at all stations before Station A could be 50 persons per minute (but may be more).

If all these people board, and the train departs the terminus 3 minutes late, then that's at least 150 extra people on board by Station A. With 950 people already, and at least 80 wishing to board, then at least 30 will be left behind. With the crush load train presenting at Stations B and C, most of their passengers will be unable to board.

Why did I say 'at least'? The graphs in Part 1 show that as a train approaches capacity, boarding time rises exponentially. So the train will likely be 5 or more minutes late at Station A, and even later down the line. And in that time more people (say 5 or 10 more per minute on average) will have passed through the validators at Stations A, B and C, all expecting to catch the train. So the train could well be full even before it reaches Station A.

Hence we have three station platforms of passengers, increasing every minute, unable to board the train. The next train, assuming it left on time might be emptier than usual on leaving the terminus, as some of its passengers will have boarded the previous late-runner. And for a while it might even be only a couple of minutes behind that train, its progress retarded by the slower-loading train in front. However as it travels more of its passengers will be comprised of those left behind than the increasingly delayed earlier train, and early arrivals for the train after that. So the second train that started on time could end up also being late and crowded.

On the busier lines in Melbourne, this and the following several trains will also be leaving passengers behind. And at the more inner stations (especially those served by one line only and bypassed by expresses) passengers may need to wait for several trains to pass before being able to board. This causes the actual delay experienced to be higher than what might be apparent from train running data alone (as some trains were unboardable).

To summarise, a line with frequent trains that are not crush-loaded (eg Belgrave/Lilydale) will outperform ones that are (Pakenham/Cranbourne and Sydenham lines) in reliability. Belgrave/Lilydale are nearer the 90% carpark full level, while the others are like 99%. And it's much easier to find 100 passener places in a hurry to clear a platform on the former than the latter.