How schedule adherence affects headways

There's an article on TheAtlanticCities which is bouncing around the office about how painful it is to wait for a train (I'd add: especially if you don't know when it might come). But even with the proliferation of countdown timers (except, uh, on the Green Line), any disruption to the published (or, at least, idealized) headways can cause headaches. And when headways get at all discombobulated, passenger loading becomes very uneven, resulting in a few very crowded trains that you, the passenger, are more likely to wind up waiting for and squeezing aboard.

For instance, let's say that you ride the Red Line in Boston. The published headway is 4.5 minutes (two lines, 9 minute headways for each line). Assuming you're going south through Cambridge, the agency should be able to send out trains at the exact headways from the two-track terminus, barring any issues on the outbound run. You'd expect that, upon entering the station, you'd have an average wait of 2:15, and the longest you'd ever wait for a train would be 4:30 (if you walked in just as the doors were closing and the train was pulling out of the station).

In a perfect world, this would be the case. In the real world, it's not. In fact, it probably seems to many commuters that their average wait for the train is more in the four-minute range, and sometimes as long as seven or eight minutes. And when a train takes eight minutes to come, the problem compounds as service bunches: the cars get too full, and dwell times increase as passengers attempt to board a sardine-can train and the operator tries to shut the doors.

Here's the rub: even if most services run on a better-than-average headway, passengers are more likely to experience a longer wait. Here's an extreme example. Imagine a half hour of service with five trips. With equal headways, one would arrive every six minutes, and the average wait time would be three minutes. Now, imagine that the first four services arrived every 2.5 minutes, and the final one arrived after 20 minutes. The average headway is still six minutes. However, the experienced average is far worse. Unless the services operate at that frequency due to load factors, passengers likely require the service at a constant (or near-constant rate). Imagine that one passenger shows up each minute. The first ten are whisked away quickly, waiting no longer than three minutes. The next 20 wait an average of 10 minutes, with some waiting as long as 20. In this case, even with the same average headway, 14 passengers—nearly half—wait longer than the longest headway if the service was evenly-spaced.

I used the Red Line as an example because I have experience with this phenomenon, and also data. Back when I first collected Longfellow Bridge data, I tracked, for two hours, how often the trains came. It turns out that the headway is actually 4:10 between 7:20 and 9:20, more frequent than advertised. However, nearly half of the trains come within three minutes, which means that there is a long tail of longer headways which pulls the average down. So instead of an average wait time of 2:05, the average user waits quite a bit longer.

Assuming that each train carries all passengers from each station (not necessarily a valid assumption), the average customer waits 2:32. This doesn't seem like a long time, but it means that while the trains are run on approximate four minute headways, the actual experience is that of five minutes, a loss of 20% of the quality of service. Five minute headways aren't bad. The issue is that there are several periods where customers wait far longer than five minutes, resulting in overcrowding on certain trains, and longer waits for the same ones. The chart below shows wait times for each minute between 7:23 and 9:23. Green is a wait of 2:15 or less, yellow 4:30 or less (the advertised headway). Orange is up to 6:45, and red is longer. About one sixth of the time a train is running outside of the given headways. And three times, it is longer than 150% of the advertised headway.

Another personal observation is that, try as I might, I seem to always get caught on a packed-full train. This is due to the same phenomenon. Of the 30 trains noted, only eight of them had headways of more than 4:30. Those 8 trains—which, assuming a constant flow of riders, accounted for 27% of the passengers—served 56 of the 120 observed minutes, carrying 47% of the ridership! Ten trains came within 2:30 of the previous trains. These trains accounted for 33% of the service, but only served 19% of the ridership. So while one-in-three trains is underloaded, you only have a one-in-five chance of getting on one of those trains. And while only about a quarter of services are packed full, you have a nearly 50% chance of riding one of those trains. So if you wonder why it always seems like your train is packed full, it's because it is. But there are just enough empty services that once a week you might find yourself in the bliss of a (relatively) empty train car.

Overall, I mean this as an observation of headways, not as an indictment of the MBTA. Running a railroad with uneven loads (especially at bus- and commuter rail-transfer stations), passengers holding doors and the like can quickly cascade in to a situation where certain trains are overloaded, and others pass by with plenty of room. Still, it's infuriating to wait. But it's interesting to have data, and to visualize what it looks like during the course of what seems to be a normal rush hour.

(On the other hand, there are some services, like the 70 bus, which have scheduled uneven headways and where the actual level of service is significantly impacted, but that's the subject of another post entirely.)

The weather is cooler. The Longfellow is the same.

Twice this summer, we counted vehicles on the Longfellow. Between June and July, when the lanes of the bridge were shifted and constricted, bicycle traffic was level (well, actually, it rose slightly) while vehicular traffic decreased. I was otherwise occupied this September and didn't get a chance to do a comparable bike count until last week, when I eked out an hour to sit on the bridge.

And the results are so mundane they aren't even worthy of charts and graphics. Basically, the numbers were within a thin margin of error of those from July:

(All values for peak-hour of the count, note that the Longfellow runs east-west; Eastbound towards Boston, Westbound towards Cambridge)

Eastbound Bikes: 308 (July: 298)
Westbound Bikes: 63 (July: 68)
Eastbound Pedestrians: 65 (July: 83)
Westbound Pedestrians: 191 (July: 201)
Inbound Vehicles: 411 (July: 415)

So the bridge, even after two months of people getting used to the traffic patterns, has seen no major changes. Any drop in non-motorized use might be attributable to cooler weather (in the mid 50s rather than the upper 60s) or to random variance. And assuming a normal traffic day, there has been no significant increase in traffic since the bridge has opened.

It's the last piece that I find most interesting. It really speaks to the concept of "induced demand." With the wider Longfellow, we say 800 vehicles per hour traversing the bridge in June. Once the bridge was narrowed, that number fell to 400. There were weeks with dozens police directing traffic, but the number of cars very quickly hit a new equilibrium. People do not seem to need a major education campaign to figure out where to go. If the new roads are gridlocked, they'll find alternate routes. The system has not ground to a halt (although inbound at the evening rush often backs up the length of the bridge). There are too many variables to find out if people have switched to other routes or modes or just not made the trip, but traffic in the morning across the Longfellow has not been the apocalypse.

Boston's Worst Traffic Day of the Year

It's tomorrow. The Friday before Columbus Day.

I have no actual data to back this up. Only anecdotal and empirical data. (Oh, and data from the Pike, which claims it's second to the Friday before mother's day in May, but I think that might be just for the Pike without the added benefit of every road north and south of the state, too. How prescient that this article comes out right after I post this.) But here's what happens, and here's how to avoid it.

Boston sees a lot of bad traffic. In the winter, when everyone is in town and weather hits, the entire system can grind to a halt. (The worst I know of was in December 2007 when a storm hit Boston around noontime. Snow fell heavily from the onset with temperatures in the mid-20s, so roads iced over. So many people left work early to beat the weather home that the roads filled up completely and plows couldn't keep them clean. So the entire network ground to a halt until snow let up late in the evening.) But you can't really plan for that. In the summer, Boston sees epic traffic jams headed out of the city to and from vacation spots, especially getting on and off of Cape Cod (the eight hour, 25-mile backup this July 4 this year was particularly bad), although other bottlenecks in New Hampshire and Western Mass can be painstakingly slow.

But the Friday before Columbus Day Weekend is the worst. Here's why:

• It's the Friday before a long weekend. So in addition to Friday traffic, you have the masses headed on vacation, too.
• But it's a normal Friday. Of all the three-day weekends in the calendar, it's the only one that almost no one extends. So there aren't many people who get away a day early to ease the traffic.
• It's the last nice weekend of the year, for foliage and, often for weather. It's still a pleasant time to go to Cape Cod, or the Berkshires, or Northern New England before the leaves fall and the temperatures plummet.
• Not many people stay in town for the weekend. On Patriots Day (Marathon), July 4 (Fireworks), Labor Day, MLK Day and Memorial Day there are parades and ceremonies and the like that people attend locally. No one is celebrating Columbus anymore.
• Oh, yeah: everyone from New York, Connecticut and Rhode Island wants to get to Vermont, New Hampshire and Maine. So guess where they all go? Massachusetts.

Personally, I've sat for two hours to go 10 miles on 128 on this wonderful day. I've also had a four hour drive from Springfield to Boston on the Turnpike. And my uncle had a three-hour trip from Providence to Boston, which culminated with him blindly following directions off of the Southeast Expressway on to Mass Ave when informed the Expressway wasn't moving.

These weren't due to accidents, but to volume. The system operates at-or-near capacity on a normal day. Add the factors above, and it is pushed way over capacity. Once that happens, everything stops.

But there are some suggestions. First of all, go where you are going later. We have wonderful apps and data and the ability to look at a computer screen and find out how long a trip is going to take. Take a look at Google Maps, or at MassDOT's traffic map or data stream, and wait it out. As long as you plan to wait it out, you can sit by the river or go for a run and wait until the coast is clear (which should happen by 7 or 8 p.m.). Second, consider back roads, especially further from the city. Much of the congestion comes tourist-types descending on to main highways. People who don't normally drive the roads don't know about parallel options, and people who are unprepared for their onslaught get caught up in the hubbub. So if the Pike is a royal mess, try Route 9. If 93 is a parking lot, come through the city. 

The saying goes "if you can't beat 'em, join 'em." But that's stupid. If you can't beat 'em, either wait patiently, or find a route where they aren't. Because if you don't, you too will get to enjoy the Worst Traffic Day of the Year!