Robert Jackel asked me an excellent question in comments: what is a pulse? I’ve talked about timed transfers a lot in the last almost 10 years of this blog, but I never wrote a precise definition. This is a critical tool for every public transportation operation with more than one line, making sure that trains and buses connect with as short a transfer window as possible given other constraints. Moreover, pulse-oriented thinking is to plan capital investment and operations to avoid constraints that make transfers inconvenient.
When are pulses needed?
Passengers perceive the disutility of a minute spent transferring to be more than that of a minute spent on a moving vehicle. This is called the transfer penalty and is usually expressed as a factor, which varies greatly within the literature. In a post from 2011 I quoted a since-linkrotted thesis with pointers to Boston and Houston’s numbers, and in a more recent post I found some additional literature in a larger variety of places, mostly in the US but also the Netherlands. The number 2 is somewhere in the middle, so let’s go with this.
Observe that the transfer penalty measured in minutes and not in a factor is, naturally, larger when service runs less frequently. With a factor of 2, it is on average equal to the headway, which is why it is likely the number is 2 – it represents actual time in the worst case scenario. The upshot is that the value of an untimed transfer is higher the higher the frequency is.
I used the principle of untimed transfers and frequency to explain why small subway networks do not look like small bus networks – they have fewer, more frequent lines. Subway lines that run every 3-4 minutes do not need transfer timing, because the time cost of an untimed transfer is small compared to the likely overall trip time, which is typically in the 15-30 minute range. But the lower the frequency, the more important it is to time transfers. Thus, for example, Berlin times the U6/U7 transfer at Mehringdamm in the evening, when trains run every 10 minutes, but does not do so consistently in the daytime, when they run every 5.
But note: while the value of an untimed transfer is higher at higher frequency, the value of a timed transfer is the same – it is zero-penalty or close to it no matter what. So really, the relative value of timing the transfer decreases as frequency increases. But at the same time, if frequency is higher, then more passengers are riding your service, which justifies more investment to try to time the transfer. The German-speaking planning tradition is the most concerned with transfer timing, and here, it is done commonly at 10 minutes, occasionally at 5 minutes, and never that I know of at higher frequency.
Easy mode: one central station
If all your buses and trains serve one transit center, then a pulse means that they all run at the same frequency, and all meet at the center at the same time. This doesn’t usually happen on urban rail networks – a multi-line urban rail system exists in a high-ridership, high-frequency context, in which the value of serving a mesh of city center lines is high, and the cost of bringing every subway tunnel to one location is high. Instead, this happens on buses and on legacy regional rail networks.
The pulse can be done at any frequency, but probably the most common is hourly. This is routine in small American towns with last-resort bus networks serving people too poor or disabled to drive. Two and a half years ago a few of us on Transit Twitter did a redesign-by-Twitter of the Sioux City bus network, which has ten bus routes running hourly, all pulsing in city center with timed connections. A similar network often underlies the night buses of a larger city that, in the daytime, has a more complete public transport network, such as Vancouver.
Even here, planners should keep two delicate points in mind. First, on buses in mixed traffic, there is an upper limit to the frequency that can be timetabled reliably. The limit depends on details of the street network – Jarrett Walker is skeptical that timetabling buses that run every 15 minutes is feasible in a typical American city, but Vancouver, with no freeways within a city and a rich arterial grid, manages to do so every 12 minutes on 4th Avenue. A half-hourly pulse is definitely possible, and even Jarrett writes those into his bus redesigns sometimes; a 20-minute pulse is probably feasible as well even in a typical American city. The current practice of hourly service is not good, and, as I point out in the Sioux City post, involves slow, meandering bus routes.
The second point is that once the takt is chosen, say half an hour, the length of each roundtrip had better be an integer multiple of the takt, including a minimal turnaround time. If a train needs 5 minutes to turn, and runs half-hourly, then good times for a one-way trip from city center are 10, 25, 40, 55 minutes; if there is no turnaround at city center, for example if there is through-running, then half as many turnarounds are needed. This means that short- and long-term planning should emphasize creating routes with good trip times. On a bus, this means straightening meanders as needed, and either extending the outer end or cutting it short. On a train, this means speedup treatments to run as fast as necessary, or, if the train has a lot of spare time, opening additional infill stops.
The issue of branching
Branches and pulses don’t mix well. The ideal way to run a system with a trunk and branches is to space the branches evenly. The Berlin S-Bahn runs every 3-4 minute on the Stadtbahn trunk and on the North-South Tunnel, mixing services that run every 10 and 20 minutes at roughly even intervals. In such an environment, timed transfers in city center are impossible. This is of course not a problem given Stadtbahn headways, but becomes serious if frequency is sparser. A one-trunk, two-branch regional rail system’s planners may be tempted to run each branch every half hour and interpolate the schedules to create a 15-minute headway on the trunk, but if there’s a half-hourly pulse, then only one branch can participate in it.
This is visible when one compares S-Bahn and RegionalBahn systems. High-frequency S-Bahn systems don’t use timed transfers in city center, because there is no need. I can get from Jannowitzbrücke to Ostkreuz without consulting a schedule, and I would get to the Ring without consulting a schedule either, so there is no need to time the crossing at Ostkreuz. There may be sporadic transfer timing for individual branches, such as between the S9 branch of the Stadtbahn, which diverts southeast without serving Ostkreuz, and the Ring, but S9 runs every 20 minutes, and this is not a pulse, only a single-direction timed connection.
In contrast, RegionalBahn systems, running at longer ranges and lower frequencies, often tend toward timed transfers throughout. The tradeoff is that they don’t overlie to create high-frequency trunks. In some cases, trains on a shared trunk may even platoon, so that all can make the same timed transfer, if high trunk frequency is not desired; this is how intercity trains are run on the Olten-Bern line, with four trains to a platoon every 30 minutes.
Medium mode: dendritic networks
A harder case than the single pulse is the dendritic network. This means that there is a central pulse point, and also secondary pulse points each acting as a local center. All cases I am aware of involve a mainline rail network, which could be S-Bahn rather than RegionalBahn, and then bus connections at suburban stations.
Already, this involves more complex planning. The reason is that the bus pulse at a suburban station must be timed with trains in both directions. Even if planners only care about connections between the suburban buses and trains toward city center, the pulse has to time with inbound trains for passengers riding from the suburban buses to the city and with outbound trains for passengers riding from the city to the buses. This, in turn, means that the trains in both directions must arrive at the station at approximately the same time. A few minutes of leeway are acceptable, since the buses turn at city center so the connection always has a few minutes of slack, but only a few minutes out of what is often a half-hourly takt.
Trains that run on a takt only meet every interval equal to half the takt. Thus, if trains run half-hourly, they can only have suburban pulses every 15 minutes of travel. This requires planners to set up suburban pulses at the correct interval, and speed up or sometimes slow down the trains if the time between suburban nodes. Here is an example I’ve worked on for a Boston-Worcester commuter train, with pulses in both Framingham and Worcester.
Hard mode: meshes
The next step beyond the dendritic network is the multi-node network whose graph is not simply connected. In such a network, every node must have a timed transfer, which imposes considerable planning constraints. Optimizing such a network is an active topic of research in operations and transportation in European academia.
Positive examples for such networks come from Switzerland. Large capital investments are unavoidable, because there’s always going to be some line that’s slower than it needs to be. The key here is that, as with dendritic networks, nodes must be located at consistent intervals, equal to multiples of half the headway, and usually the entire headway. To make multiple timed transfers, trains must usually be sped up. This is why pulse-based integrated timed transfer networks require considerable planning resources: planning for rolling stock, infrastructure, and the timetable must be integrated (“the magic triangle”) to provide maximum convenience for passengers connecting from anywhere to anywhere.