The Waiting Time Paradox, Or, Why Is My Bus Always Late?

I don't think this is always true. My city measures buses as being on time at inner stops as being between something like 5 minutes early and 1 minute late (yes, you read that right). Timing stops, where they have to wait, are pretty infrequent (mostly bus terminals).

I don't know how common it is but it does exist. And buses perpetually being early means that if you're on time you wait even longer for the next one.

This is pretty counterintuitive. In the game described you should expect to see one 'wrong way' bus per play on average, not half as you might expect. On the other hand, you have an exactly even chance of catching your own bus before seeing a wrong one, so if your scoring system had been +1 for your bus and -1 for one or more wrong ones, then you would indeed score 0 over time. But with your point per bus scoring system your expected score turns out to be -0.5 per play.

It's all much easier than that:

It's just the Poisson process, e.g., with a nice chapter in E. Cinlar, Introduction to Stochastic Processes.

Buses come as arrivals. So bus arrivals are a stochastic arrival process where stochastic just means varying randomly over time where, really, the randomly doesn't mean anything, includes deterministic arrivals, that is, known exactly in advance, but also admits any case of unpredictability.

Well, in short, if have a stochastic arrival process with stationary, independent increments, then the arrival process is a Poisson process and there is a number, usually denoted by lambda, so that the times between arrivals are independent, identically distributed random variables with exponential distribution with arrival parameter, the arrival rate, lambda. The stationary means that the probability distribution of the times between arrival does not change over time. The independent increments means that the time from one arrival to the next is independent of all the past history of arrivals.

The exponential distribution has the property, easy to verify with simple calculus, that the conditional expectation of the arrival time given that the arrival time is already greater than some number is the same as the expected arrival time.

So, net, if bus arrivals form a Poisson process, then the time until the next bus arrives is the same after waiting five minutes as not having waited at all.

Cinlar's treatment is nice because it is qualitative, that is, has assumptions that can often be confirmed or believed just intuitively. And we might not believe that bus arrivals meed the assumptions.

This subject can continue with, say, hazard curves for equipment failures and a lot more about Poisson processes.

E.g., the sum of two independent Poisson processes, say, Red buses and Blue buses, assuming that they are Poisson processes, is also a Poisson process with arrival rate the sum of the Red and Blue arrival rates. If randomly throw away some arrivals, then what is left is also a Poisson process with arrival rate adjusted in the obvious way.

In Feller's volume II is the renewal theorem that the sum of independent arrival processes, Poisson or not, with mild assumptions, converges to a Poisson process as the number of processes summed grows. So, if the users of a sufficiently busy Web site act independently with mild assumptions, then the Web site will see arrivals accurately as a Poisson process.

The vanilla Poisson process is Geiger counter clicks.

There is much more to the pure and applied math and applications of Poisson processes.

> What is an actual phenomenon that is well modeled by a poisson process?

Geiger counter clicks.

Arrivals between, say, 2-3 PM at a busy Web site.

Equipment failures for equipment with a constant hazard curve -- if can find such equipment.

Time between road kills on a highway.

This reminds me of a mathematical paradox that makes me doubt your conclusion: "In this country, every couple wants to have one daughter. They keep having children until they have a daughter, and then they stop. What gender balance should we expect?"

Couples can have any number of sons, and every couple has exactly one daughter. Still, the accepted mathematical solution is an equal gender ratio for the couples' children.