Frequency in Units of Distance
I have annoying commenters. They nitpick what I say and point out errors in my thinking – or if there are no errors, they take it beyond where I thought it could be taken and find new ways of looking at it. After I wrote about frequency relative to trip length last week, Colin Parker pointed out on the Fediverse that this can be simplified into thinking about frequency not in units of time (trains or buses per hour), but in units of distance (trains or buses per km of route). This post is dedicated to developing this idea on various kinds of transit service, including buses and trains.
The key unit throughout, as Colin points out, is the number of buses available per route, the assumption being that the average trip length is proportional to the average route length. However, this is not a perfect assumption, because then the introduction of network effects changes things – generally in the direction of shorter average trip length, as passengers are likelier to transfer, in turn forcing agencies to run more vehicles on a given route to remain useful. Conversely, timed transfers permit running fewer vehicles, or by the same token more routes with the same resources – but the network had better have a strong node to connect to after a series of vehicle changes, more like the Swiss rail network than like a small American city’s bus network.
Frequency and resources
On a bus network with even frequency across all routes, the following formula governs frequency, as I discussed six years ago:
Daily service hours * average speed per hour = daily trips * network length
When Eric and I proposed our Brooklyn bus redesign, we were working with a service-hour budget of about 10,800 per weekday; status quo as of 2017-8 was 11 km/h, 550 km, and thus 216 daily trips (108 per direction), averaging around a bus per 11 minutes during the daytime, while we were proposing speed up treatments and a redesign to change these figures to 15 km/h, 355 km, and thus 456 daily trips (228 per direction). The six-minute service ideal over 16 hours requires 188 trips per direction; the difference between 188 and 228 is due to higher frequencies on the busiest routes, which need the capacity.
To express this in units of length, we essentially eliminate time from the above dimensional analysis. Daily service hours is a dimensionless quantity: 10,800 hours per weekday means 450 buses circulating at a given time on average, in practice about 570 during the daytime but not many more than 100 buses circulating overnight. If there are 570 buses circulating at a given time, then a 550 km network will average a bus every 1.9 km and a 355 km one will average a bus every 1.25 km. With pre-corona New York bus trips averaging 3.4 km unlinked, a bus every 1.9 km means the maximum headway is a little higher than half the trip time, and a bus every 1.25 km means the maximum headway is a little higher than one third the trip time, independently of speed.
This calculation already illustrates one consequence of looking at frequency in units of distance and not time: your city probably needs to aggressively prune its bus network to limit the wait times relative to overall trip times.
Route length and trip length
On an isolated bus or train route, serving an idealized geography with a destination at its center and isotropic origins along the line, the average trip length is exactly one quarter of the route length. The frequency of service in units of distance should therefore be one eighth of the route length, requiring 16 vehicles to run service plus spares and turnarounds. This is around 18-20 vehicles in isolation, though bear in mind, the 10,800 service hours/day figure for Brooklyn buses above is only for revenue service, and thus already incorporates the margin for turnaround times and deadheads.
Colin points out that where he lives, in Atlanta, bus routes usually have around four vehicles circulating per route at a given time, rather than 16. With the above assumptions, this means that the average wait is twice the average trip time, which goes a long way to explaining why Atlanta’s bus service quality is so poor.
But then, different assumptions of how people travel can reduce the number 16:
- If destinations are isotropic, then the average trip length rises from one quarter of the route length to 3/8 of the route length, and then the frequency should be 1.5/8 of the route length, which requires 11 vehicles in revenue service.
- If origins are not isotropic, then the average trip length can rise or fall, depending on whether they are likelier to be farther out or closer in. A natural density gradient means origins are disproportionately closer-in, but then in a city with a natural density gradient and only four buses to spare per route, the route is likely to be cut well short of the end of the built-up area. If the end of the route is chosen to be a high-density anchor, then the origins relative to the route itself may be disproportionately farther out. In the limiting case, in which the average trip is half the route length, only eight buses are needed to circulate.
To be clear, this is for a two-tailed route; a one-tailed route, connecting city center at one end to outlying areas at the other, needs half the bus service, but then a city needs twice as many such routes for its network.
The impact of transfers
Transfers can either reduce the required amount of service for it to be worth running or increase it, depending on type. The general rule is that untimed transfers occurring at many points along the line reduce the average unlinked trip and therefore force the city to run more service, while timed transfers occurring at a central node lengthen the effective trip relative to the wait time and therefore permit the city to run less service. In practice, this describes both how existing bus practices work in North America, and even why the Swiss rail network is so enamored with timed connections.
To the point of untimed transfers, their benefit is that there can be very many of them. On an idealized grid – let’s call it Toronto, or maybe Vancouver, or maybe Chicago – every grid corner is a transfer point between an east-west and a north-south route, and passengers can get from anywhere to anywhere. But then they have to wait multiple times; in transit usage statistics, this is seen in low average unlinked trip lengths. New York, as mentioned above, averages 3.4 km bus trips, with a network heavily based on bus-subway transfers; Chicago averages a not much higher 3.9 km. This can sort of work for New York with its okay if not great relative frequency, and I think also for Chicago; Vancouver proper (not so much its suburbs) and Toronto have especially strong all-day frequencies. But weaker transit networks can’t do this – the transfers can still exist but are too onerous. For example, Los Angeles has about the same total bus resources as Chicago but has to spread them across a much larger network, with longer average trip times to boot, and is not meaningfully competitive. The untimed grid, then, is a good feature for transit cities, which have the resources and demand to support the required frequencies.
Not for nothing, rapid transit networks love untimed transfers, and often actively prefer to spread them across multiple stations, to avoid overwhelming the transfer corridors. Subways are only built on routes that are strong enough to have many vehicles circulating, to the point that all but the shortest trips have low ratios of wait to in-vehicle times. They are also usually radial, aiming to get passengers to connect between any pair of stations with just one transfer; Berlin, Paris, and New York are among the main exceptions. These features make untimed transfers tolerable, in ways they aren’t on weaker systems; not for nothing, a city with enough resources for a 100 km bus network and nothing else does not mimic a 100 km subway network.
Timed transfers have the opposite effect as untimed transfers. By definition, a timed transfer means the wait is designed to be very short, ideally zero. At this point, the unlinked trip length ceases to be meaningful – the quantity that should be compared with frequency is the entire trip with all timed transfers included. In particular, lower frequencies may be justifiable, because passengers travel to much more than just the single bus or rail route.
This can be seen in small-city American bus networks, or some night bus networks, albeit not with good quality. It can be seen much more so on transfer-based rail networks like Switzerland’s. The idealized timed transfer network comprises many routes all converging on one node where they are timed to arrive and depart simultaneously, with very short transfers; this is called a knot in German transit planning and a pulse in American transit planning. American networks like this typically run a single bus circulating on each one-tailed route; the average wait works out to be four times longer than the average unlinked trip, and still twice as long as a transfer trip, which helps explain why ridership on such networks is a rounding error, and this system is only used for last-resort transportation in small cities where transit is little more than a soup kitchen or on night bus networks that are hardly more ridden. It would be better to redo such networks, pruning weaker routes to run more service on stronger ones, at least two per one-tailed route and ideally more.
But then the Swiss rail network is very effective, even though it’s based on a similar principle: there’s no way to fill more frequent trains than one every hour to many outlying towns, and even what are midsize cities by Swiss standards can’t support more than a train every half hour, so that many routes have a service offer of two to four vehicles circulating at a given time. However, on this network, the timed transfers are more complex than the idealized pulse – there are many knots with pulses, and they work to connect people to much bigger destinations than could be done with sporadic one-seat rides. A succession of timed connections can get one from a small town in eastern Switzerland to St. Gallen, then Zurich, then Basel, stretching the effective trip to hours, and making the hourly base frequency relatively tolerable. The key feature is that the timed transfers work because while individual links are weak enough to need them, there are some major nodes that they can connect to, often far away from the towns that make the most use of the knot system.
Pedestrian Observations 
I impatiently wait for the post where you describe the Doppler effect of a pedestrian walking along a transit route.
That’s why there are red and blue lines…
And when the buses get stuck in congestion to an average speed below ~5 km/h, the pedestrians cause Pokey radiation.
This is more complex in Heidelberg, where you can never be quite sure where the bus is at any given moment in time….
And in Kaliningrad (Kralovets?), where no matter how they try to plan the routes, all the buses always end up stuck on the wrong side of the river, or on some island.
I tried for a few minutes to find a city that sounded like Schrödinger to point out how they have trouble tracking ridership because they don’t know if the passenger is inside the bus or not, but couldn’t find anything close enough.
The Mobius bus always get you where you are going and never has to turn around …
Lobachevsky was a transit planner that was tasked with a bus grid in a city without parallel streets
Engaging nit picking node, isn’t your formula dimensionally inequivalent with the units of km on the left and trip.km on the right?
What is the local trip situation like in Switzerland with hourly knots? I’d expect to see most people driving in small towns for their local trips, but when they “got to the city” the train starts to look a lot nicer and then they use local transit instead of trying to park at the train station. Or does the existence of a strong place to get to (via train) induce locals to put up with hourly service for local trips. This might be partially an economics question – I could believe that those towns are poorer and so people can’t really afford a car, but the existence of transit that can go places mean they can afford to live there instead of being forced to move to a richer city.
Based on my relatives in Switzerland they all have cars but most also do significant transit trips. So the choice depends on the destination. This seems to hold whether they are in major cities or small towns
For what I understand, Swiss towns will either be small and dense enough to walk for most local trips, or will have excellent local transit as provided by trolleybus or tram if you don’t want to take your car. Nonetheless, Switzerland is an extremely rich country with high rates of car ownership, so high ridership of both local and intercity transit speaks to doing something right.
The big win of public transport in Europe is that fewer people have second cars.
Even though it is extremely rich Switzerland has 604 cars per 1000 people vs 908 in the United States. This is as per https://en.wikipedia.org/wiki/List_of_countries_and_territories_by_motor_vehicles_per_capita#
Sorry to be that person, but don’t some American cities do do pulses well? Here in Massachusetts BAT in particular comes to mind.
To clarify, the difference between a “good” pulse and a “bad” pulse is the frequency at the connection itself? Or are there other factors? I know from experience that having one route be off the pulse for whatever reason in this case does feel like it does more damage, but that’s not really a problem with the pulse itself.
The pulses are executed well in the sense of having timed connections. But everything else about them makes them fail. BAT is a really good example for this, and I say this as someone who looked into it as part of the TransitMatters Regional Rail proposal. BAT’s problems include,
1. Total lack of integration with commuter rail in schedules, fares, or who is supposed to ride the vehicles. There’s even a route that parallels the Middleborough Line going to Ashmont, because commuter rail is too expensive and infrequent.
2. Irregular frequencies – all buses run on the same frequency and connect in Brockton, right next to the train station, but the frequency is not the same all day.
3. Insufficient span – buses serve malls, with a clientele comprising retail workers, but the last bus of the day leaves hours before the mall closes, making it useless for the only people poor enough to need it.
Point 3 is a general problem with American transit; point 1 is what motivates my comparison with the Swiss Takt – if those buses ran every half hour and pulsed with commuter rail, and if commuter rail had fare integration so that BAT + commuter rail cost the same as BAT + subway does today, then those half-hour headways would be smaller not just in absolute terms but also in relative terms to the trip length, because the trip might well be from an outlying point around Brockton to a point in or near Downtown Boston.
This is actually why I prefer train shuttles over buses. Specifically, many Caltrain stations have shuttles that are timed with baby bullet train arrivals and departures, so as soon as you get off a train, you can get on a shuttle that will depart within a few minutes.
Secondly, the shuttles are free, making the trip costs simple.
It’s rare for the bus schedule to be driven by train arrivals and departures. They typically run at intervals, meaning the are timed well for some trains, but frequently are not.
It’s also rare for a bus to WAIT for a delayed train. If your train is late, you’re delayed by 15-20 mins for the next one.
It’s rare for buses to begin their trips at the station. Stations are frequently an intermediate stop, meaning you need to stand around waiting for the bus.
Buses that DO originate, rarely allow passengers to board to wait for a departure. Bus drivers are frequently on a break and refuse to provide access to the bus.
Fortunately, Caltrain has many free shuttles at majority of the stations:
https://www.caltrain.com/schedules/shuttles
Why are the trains not running on a Takt that the buses can meet with their intervals? (I guess there’s an hourly local Takt and an hourly express Takt off-peak, but that’s weak for a line of this length.)
The station is served by local, limited, and express trains, so while there’s a takt for each service level, trains arrive at different times.
The shuttle service is optimized for express trains that arrive from both directions, so it’s tricky. Since some routes are served by one vehicle, not every train is serviced, and some times work better by being able to serve more trains.
If you want to use regular buses, you’d need to make the following changes:
Your authoritarian streak is showing. Again. Tell us Dear Leader, who should be riding the vehicles? And what should be done if the proletariat is insufficiently motivated?
It’s awful that they aren’t running a bus every half hour 24 hours a day, to spare you the trouble of looking at a schedule. They have many accommodations your aversion to timetables/schedules.
They let retail workers buy cars. Malls without bus service get staffed. People who want to take the bus to work, schedule a shift when the buses are running.
I glanced at the schedules. And noticed the frequency, on routes that have half hourly frequency, go up to half hourly during regular business hours. When people who work regular business hours are working. And unavailable for shopping trips.
Like every other non-urban bus system you misinterpret this is for the few people who cannot drive. There is enough free parking that people who have access to a car will drive.
“who is supposed to ride the vehicles” means who is expected to ride. And that is overwhelmingly lower income people such as mall workers, for whom owning a car is a huge financial burden, if they can afford it at all.
The rest of your comment is incoherent (e.g. “half hourly frequency, go up to half hourly during regular business hours”)
How upper middle class puritanical of you. Mall workers can do whatever they want with their money. And just like they let mall workers buy cars they let rich people ride the bus. Like walk to the bus line a block away instead of hiking to the rail line that Alon thinks is parallel.
Alon whined that the buses don’t have the same frequency all day long. So I actually looked at the schedules. They don’t. Silly silly bus system running fewer buses when most people are still asleep. A few of them go from an hourly schedule to half hourly. During regular business hours when people who work regular business hours are already in work.
Employers don’t pay retail workers enough for them to buy cars and afford other expenses, so no, the American system doesn’t really let retail workers buy cars, and simultaneously doesn’t have good enough public transit for them to have an alternative. They end up coping by having one car for a multi-person household, with jockeying for who gets the car at peak times, and this is also probably an old, unreliable car with high maintenance costs (it’s expensive to be poor in the US).
That’s you deciding on how they are going to spend their money. Again.
Explain all the retail, that doesn’t have bus service, with staff. The view from your ivory tower, across Ninth Avenue, is blurry.
“your city probably needs to aggressively prune its bus network to limit the wait times relative to overall trip times.”
Maybe I’m stupid, but can you explain to me why “wait times relative to overall trip times” matters? If overall trip time is low, shouldn’t the service be competitive and draw riders regardless what fraction of that is wait time? (I know in your previous post you argue that wait time is counted 2x compared to travel time, but for a low enough overall trip time both wait and travel time are low.) What is the goal here – maximizing ridership for a given budget, maximizing human wellbeing, maximizing profitability, or something else?
I think in terms of maximizing ridership, usually.
The issue with looking (just) at overall trip time is that in the cases in question, if it’s low, then so is the car travel time. A 15-minute bus ride is not competing with cars that take 30 minutes but with cars that take 10 minutes, so if the bus ride has a 30-minute wait tacked on top of it, you’re not getting above-poverty ridership.
Can you explain the mathematical calculations used to achieve the kilometres per bus measure? I’m not understanding how from a 355 or 550km bus network with 570 buses you get a bus every 1.25 or 1.9km, the division is not working out for me so I must be missing something.
There’s an extra factor of two coming from the fact that there are two directions, whereas the 355 and 550 km figures are bidirectional (so a 10 km two-way bus route is counted as 10 km, not 20).
Because the the station is serviced by express, express, and local train, so while those might be takt based, the shuttles are aligned with the express trains.
The shuttles also try to hit both northbound and southbound trains, so the wait time is not always the same.
The shuttle service runs around hourly or maybe every 30 mins with the same vehicle to match the service.
So yeah, maybe you could do buses on a takt, but the first problem is that you won’t be able to justify high service level to make up for non-takt train arrivals.
You’ll also need to convince operators to provide it for free for train riders.
You’ll also need to convince people that buses should hold departure for train riders.
You’ll also need to convince shuttle riders to deal with slower service because the bus will also be picking up riders and collecting fare instead of just dropping them off.
In the end, you’ll probably end up with better bus service that keeps train riders away by providing slower and worse service to anyone transferring.
From experience, the shuttles were very well utilized, so they were doing something right.
Martin, your entire personal context appears to be “hopelessly bad San Francisco–San Jose train `service’ and catastrophically shit bus `service’ in the period 1995-2024.”
That may indeedv be valid as your personal lived experience of the world, but … consider this, just for a few fleeting moments, consider this … your personal experience may have been (through no fault of your own! Due solely to global-cultural-historical circumstance!) utterly shit. Utterly, totally shit.
This is the “experience” of profound Stockholm Syndrome, of somebody who simply has no conception, no conception at all, of what a barely functioning, barely integrated, barely regional transportation network might even look like.
It’s adorable that you can come up, all by yourself, with “reasons” and “problems” that mean that “operators” and “people” should actively seek out the shittiest experiences. There can be no question, no question at all, at that Silicon Valley, California, U. S. A., U. S. A., U. S. A., in this year of our Lord 2024, in the Undisputed Global Thought Leadership Think Tank Incubator of the planet, is providing the very very best possible “bus service that keeps train riders […] while providing [faster] and [better] service to anyone transferring.”
Those “shuttle riders” will indeed need no convincing that they are not enjoying the best and fastest possible service. I mean, they live in the richest region of the planet in the most off-the-scales affluent period of eukaryote history, right? Their “connecting” bus service must surely be the finest that any bilateral vertebrate might ever hope to experience, right? Right? Right?!?
I mean honestly if it at least tries to meet the train that is an improvement over the Oxenholme-Windermere-Ambleside service in the UK.
It’s basically Swiss level at that point.
Matthew Hutton: I know of a branch line that was for a few years timetabled to miss the intercity connection by a few minutes. The service planners had not the internal-political clout to tell the infrastructure side to do anything particular, such as to spend the epsilon money to install spring-loaded points, even though this would also allow the station to be unstaffed. Nor could they insist on double-staffing it, thus a single signalman had to run between both ends of the station (and the box) carrying points “keys” (these happened to be more or less what you’d call ground frame points; thrown by a lever next to them) back and forth. The union insisted on the standard norm time for this, despite crossings only happening once per two hours, and the line having zero freight traffic.
And in this anti-Switzerland, the national long-distance bus company (Volánbusz) had been merged with the railway company (MÁV), yet there is no cooperative scheduling, nor ticketing/fare integration, between the services run by the two departments of the one company.
Rather than wait for US to become Switzerland, I applaud small wins like these that make taking transit possible. We could wait for fares to be integrated, for frequency to improve, for schedules to be improved, for more buses to be purchased, for more drivers to be hired, and for more tax-money to arrive to pay for all that.
I’d happily choose an efficient option that gives me all of the above, even if it means adjusting to an hourly schedule. Takt is great, but there are so many cases were just a little bit of planning, can give you a faster trip.