Street grids, transit headways, and queueing theory. I moved to a city with a bad transit system a few years ago and started reading about why, and the short version is that every complaint anyone has about a city is a complaint about a queue.

Little's law, applied to a bus

The most useful equation in the field is also the shortest. Little's law says that for any stable system, the average number of things inside it equals the arrival rate times the average time each thing spends inside.

L = λW

It does not care what the things are. Packets, customers, cars, people on a platform. If a subway station has passengers arriving at 200 per minute and each spends an average of 4 minutes inside, there are 800 people in the station. Always. You cannot argue with it. If the platform holds 600, you have a problem that no amount of signage will fix, and the only levers are λ (fewer people arrive) or W (they leave faster).

W is mostly waiting. And waiting is where headways come in.

The ten-minute cliff

A headway is the time between successive vehicles on a route. There is a threshold, somewhere around 10 to 12 minutes, below which people stop reading the schedule. They just go to the stop. Above it, they plan their departure around the timetable, which means a late bus costs them the entire headway, not the lateness.

The average wait for a randomly arriving passenger on a route with regular headway h is h/2. At a 30-minute headway that is 15 minutes of pure W, per trip, before the vehicle moves. At 5 minutes it is 2.5. The difference in service isn't the ride. It's the 12.5 minutes you didn't spend standing on a corner, multiplied by every rider, every day.

This is why frequency, not speed, is the variable that makes a transit line feel usable. A bus that goes 20% faster saves you 3 minutes on a 15-minute ride. A bus that comes twice as often saves you 7.5 minutes of waiting, and lets you stop checking the app, which is a different kind of cost that nobody puts in the model but everybody feels.

And there is a nasty nonlinearity. Buses bunch. A bus that falls slightly behind picks up more passengers at each stop, because more have accumulated, which makes it fall further behind, while the bus behind it finds empty stops and catches up. Uncorrected, a 10-minute headway degrades into two buses arriving together every 20 minutes. Positive feedback loop, no equilibrium, and the average wait doubles without a single bus being removed from service.

Roads: the demand you build

Cars get their own queueing pathology, and it's worse.

Duranton and Turner, 2011, looked at US metro areas across decades and found that vehicle miles travelled increase roughly one-for-one with lane miles built. Elasticity close to 1. Add 10% more highway, get 10% more driving, congestion back where it was. They called it the fundamental law of road congestion. Engineers had suspected it since the sixties, but this was the cleanest measurement.

The mechanism is not mysterious. Congestion is the price of road. Lower the price, and latent demand, trips people were not making because it was too slow, shows up to consume the capacity. You can't build your way out because the thing you're building is the thing that generates the demand.

Then there's Braess's paradox, which is the one that sounds fake. Adding a road to a network can make everyone's commute longer. Each driver, choosing selfishly, routes through the new shortcut, which overloads the links feeding it, and the new equilibrium is worse than the old one for every single driver. It has been observed in Stuttgart, in Seoul when they tore out the Cheonggyecheon expressway and traffic improved, and in New York when they closed 42nd Street for Earth Day and nothing happened.

Roads aren't pipes. They are a network game with a bad Nash equilibrium.

The grid is a routing table

The Manhattan grid was laid out in 1811 by three commissioners who had no idea what would go on the land and did not want to know. That was the design goal. A grid makes no assumptions about use. Every block is interchangeable, every address computable, every route a Manhattan distance, which is a term because of the grid, not the other way around.

The trade is that a grid has a lot of intersections, and intersections are where throughput dies. A road's capacity isn't the number of lanes. It is the capacity of its worst intersection, because that is where flows have to take turns. Every signal cycle, one direction waits while the other goes, and the waiting is dead time that no lane count recovers. A six-lane arterial feeding a four-way signal moves about as many cars as the signal lets through, and the extra lanes mostly serve as queue storage.

This is why the useful trick in signal timing is coordination, the "green wave" where lights along a corridor are offset so a platoon of cars moving at the design speed hits green after green. Good engineers can tune a corridor so cars almost never stop. Bad ones, or just uncoordinated ones, produce the experience of catching every red on a street where every light is on a timer, which is the city equivalent of a distributed system with no clock sync.

Traffic is a fluid with a memory

When a road is lightly loaded, cars are particles. When it saturates, they become a fluid, and the fluid has a striking property: jams move backward.

A car brakes. The one behind brakes a little harder, because reaction time. The one behind that, harder still. The disturbance propagates upstream as a compression wave at a fairly consistent speed, around 15 to 20 km/h, against the flow of traffic. Which is why you hit a jam on the freeway, crawl through it, and emerge into open road with no accident, no merge, no cause. The cause was a tap on the brakes ten minutes and three kilometers ago. The wave outlived it.

The same math describes gas dynamics and, strangely, crowds. Above a certain density the individual agents stop being the unit of analysis. You are modeling a medium.

The bottleneck is always the shared thing

The pattern under all of this, if there's one: a city's capacity is the capacity of its most contended shared resource, and adding capacity to anything else just moves the queue.

Build more lanes, the queue moves to the intersections. Fix the intersections, it moves to parking. Add trains, it moves to the platform stairs. Widen the stairs, it moves to the fare gates. At every stage the measured throughput of the whole system equals the throughput of the one component that's saturated, and every other component is running below capacity, doing nothing useful, and costing money.

This is also, exactly, how you tune a database. Find the saturated resource, relieve it, find the next one. The difference is that a city can't be paused for maintenance, its users are voters, and every one of them has a strong opinion about the schema.