Optimal stop spacing and travel distance
In my previous post on stop spacing I made the point that for short trips - particularly the last mile shortening stop spacing counter-acts the benefits of shorter walks because the transport slows down. Efficiency and speed is currently under-rated - in comparison to connectivity and frequency - in transport discussions (at least in Melbourne). It shouldn't be, for two reasons: speed is the primary determiner of the route and travel method chosen; and faster transit leads to faster turn-arounds and therefore fewer trains and drivers for the same frequency. A 20% improvement in travel speed doesn't increase capacity (that remains throughput) but it would have massive implications for recurring costs.
In this post I'll discuss trips of varying lengths, in order to make a simple but important point: in a walk-transit-walk environment optimal stop spacing is a function of travel distance. A secondary point will also be made, with caveats: that for many systems, particularly in Melbourne, stop spacing is much too close. (A point made in relation to trams in Melbourne by Jarrett Walker)
Firstly a few assumptions. Adjusting them may make some small differences - and if anyone wants the spreadsheet I did this on, just ask - but less than you might think. For the sake or argument I am assuming a grid with even density, so average walking distance is equal to stop spacing at both ends of the journey: half the users will walk less than half stop spacing, half will walk more than half, with those in the centre of the grid traversing half in both a N-S and E-W direction. The transport in question has a 1m/s2 acceleration and deceleration time, with intersections ignored (ie. light-rail, either grade separated or gated), a 40 second stop penalty, 5 minute waiting average, and a walking speed of 5km/h. Graphs will reflect averages; walking, transport and waiting time will vary, obviously.
But to emphasise, again, adjusting these numbers makes very little difference to optimal stop spacing: waiting time is a constant, and only matters if someone can walk the distance faster - ie. for very short trips. Otherwise walking and transport speed are minimised at the point where transit speed is not compromised by frequent stops: the major factor in determining optimal stop spacing is the distance being travelled on transit (given a particular walking speed).
Optimal stop spacing in a walking only environment
Below is the journey speed for a 4km trip, given the assumptions above. There are two things worth noting. Firstly, that optimal stop spacing is 850m, which is outside the generally accepted range for short-ish trips of this kind. Secondly, that the cost of sub-optimal spacing is much higher on the short side. A 400m stop spacing is of a piece with a 1700m stop spacing, and the cost of reducing it further much higher.
There is an important, and unresolved tension then, between two conceptions of walking to transit: is the commuter rational, and therefore willing to walk whatever distance affords them the fastest trip, where speed is all that matters; or are they unwilling or unable (speaking here of the general population, not the mobility impaired - and of a decent walking environment, as that can be fixed) to walk further, even if it meant faster transit? This graph from VISTA data (courtesy Alan Davies), would indicate that people are willing to walk reasonable distances for trains (which on average have longer travel distances), but may merely indicate that many train stations are further apart. Similarly, while there is a significant clustering effect around train lines in Melbourne, that is in part because the stop spacing is short, and therefore geared for short walks.
The graph below shows the optimal stop spacing for various trip lengths. The levelling out at 2km is a function of the maximum travel speed of the transit, but shows that except for very short trips the optimal stop spacing is in excess of 1km, and growing. Keeping in mind the previous point that longer is better than shorter, a large percentage of commuters in Melbourne are suffering excessively long travel times.
Optimal stop spacing with a connecting transport
While Melbourne has an high percentage of walking only access to trains, this is probably a reflection of both poor local connections, and the short stop spacing that allows extensive walk-up access at the expense of travel speed. I'll cover the relative speed of Melbourne public transport in a future post. In large cities, with long commutes, feeder systems allow for a slightly different stop spacing arrangement, because they can cover for shorter trips, and allow the train system to focus on longer trips at higher speed.
In the following graphs, a feeder system, travelling at an average speed of 30k/h with a 400m stop spacing takes passengers between stops with a 5min wait for a connecting service. The difference this makes to a 20km trip can be seen below:
Again two points are worth taking from this graph. The first is that a significantly longer stop spacing is optimal when there is a connecting service. The second is that the margin of error for the stop spacing on the long distance service is much higher. Anything from a 3km to 8km stop spacing gives a broadly similar transit speed because the service is mostly running at full speed. This is important, because transit must, of necessity, serve trips of different lengths. If the speed is broadly similar regardless of the stop spacing then (provided the basic minimum stop spacing is achieved) transit agencies can place stations in major centres to maximise connectivity.
The graph for optimal stopping distance across all commute lengths shows how much further out the stops can be with a good connecting service.
This conclusion is in some ways obvious: naturally transit can go much faster if it doesn't stop, and naturally systems that interact will work better. But it leads to several important points:
- It is better to have stop spacing too long than too short, because the time penalty is significant.
- Transit for long trips and short trips is not interchangeable. This is particularly true if transit designed for long trips has better frequencies, as it will out-compete local transit, making it redundant.
- A system that allows bleed between the roles of different transit will be sub-optimal. Part of long term planning should be to optimise the system for efficient travel by reorganising stops and connections.
Needless to say this has important implications for Melbourne's public transport system.
 Here I have assumed walking is a constant, but note that even at very slow speeds (1km/h) the optimal stop spacing for a 10km trip is 650m, compared to 1300m for a walking speed of 5km/h.
 This would be an unusually fast service given that level of stop spacing, and most likely, a street running route. Halving this to 15km/h reduces optimal stop spacing for the rapid transit service by around 15%
26th May, 2014 01:20:55
Optimal stop spacing and travel distance
This is an interesting analysis Russell, though there are two additional factors you really ought to consider as they have quite an important effect on the conclusions.
The first is that you seem to be assuming a fixed time penalty at every stop regardless of how many stops there are or how many passengers get on and off at each. In reality, if you eliminate half the stops on the route without losing half your passengers, there will be twice as many people getting on and off at each stop, and that will have a big effect on the time spent at each stop. Beyond a certain point the increased dwell times at the smaller numbers of stops will almost cancel out the time advantage from not stopping as often. This is because the fixed time penalty from making an additional stop is really just the deceleration time plus the time taken to open and close the doors: any additional time taken will vary with the number of passengers getting on and off.
The other factor to consider is that the value of people's time is not a constant. For most people in their daily routine, walking is a more 'costly' activity than being stationary in a vehicle (especially if seated). So they may well be prepared to trade off an extra minute's walk for an extra two minutes spent on board which they can use for greater benefit. (Even when we go places by car we'll try for a park close to our destination rather than blocks away, even if that means more time spent searching for one. And time on public transport can be used much more productively than when driving a car.)
It's quite true and not surprising that if you discount these factors, you find the most benefit comes from locating stops very far apart, because not only do you suffer no time penalty by concentrating arbitrary multiples of people at a small number of stops, but the people themselves find it no inconvenience to spend the greater part of their journey on foot, so as to maximise the speed of the rapid transit service.
But you can make different assumptions, and these can change your conclusions dramatically. The PTUA's own (contestable) assumptions and conclusions are here, from an exercise done some years ago:
Tony Morton 26th May, 2014 18:02:50
Optimal stop spacing and travel distance
Tony, thanks for that. As we discussed on twitter (and as I hope was made reasonably clear by the distances talked about) we are really talking about trains, not trams, with the potential for higher speeds in their own right-of-way.
You are correct that I posited a fixed dwell time. However, for trains, a large portion of the time losses that matter (on the upward slope of stops too close together) come from losing the opportunity to travel at high speed. I agree with the PTUA conclusions with regard to trams, and the numbers are broadly in line with those in my previous post.
With regard walking the greater concern is that people just won't walk that far. The societal benefits if they did, in addition to the efficiency benefits for the system would be substantial. Much greater than the value of an extra minute sitting down - assuming one can find a seat! I should note too, that this post (and the last) was meant to be largely theoretical, and posits a commuter rationalising their time cost. In that sense, the second half is more important, I think, because it shows the value of combining short stop spacing on local trips with connecting rapid services. Also, as I've (hopefully) shown, that optimal stop spacing depends on trip length, there can be no optimal stop spacing, as (by definition) there is no fixed trip length. But hopefully that also shows the folly of trying to do too much with one mode - something Melbourne struggles with.
I agree with your general point though, that it comes down to how big a benefit can be achieved from adjusting stop spacing and connecting services. For local transport, the answer is not very much (particularly in the CBD itself). For the train system, I suspect, but will try and quantify, that the answer is quite a lot.
Russ 26th May, 2014 21:16:25