24 October 2026

Why the queue at the bank is not Poisson

Date: 24 October 2026

A closed form is a reason to choose a model, and that is the wrong reason.

The arrival process at a bank on the first of the month is not Poisson. Everybody who has ever built the system knows this. The model in the slide deck still says Poisson, because Poisson is the one with a closed form, and a slide with a closed form is a slide that can be finished.

This is not a story about a lazy analyst. It is a story about what a closed form does to a decision.

What Poisson buys you

An M/M/c queue is solvable. If arrivals are Poisson with rate λ\lambda and service is exponential with rate μ\mu, the probability that an arrival waits at all is the Erlang C formula:

C(c,a)=acc!⋅cc−a∑k=0c−1akk!+acc!⋅cc−aC(c, a) = \frac{\dfrac{a^c}{c!}\cdot\dfrac{c}{c-a}}{\displaystyle\sum_{k=0}^{c-1}\frac{a^k}{k!} + \dfrac{a^c}{c!}\cdot\dfrac{c}{c-a}}

where a=λ/μa = \lambda/\mu is the offered load. Every quantity a manager asks for — mean wait, probability of waiting, the number of tellers needed to hold the wait under four minutes — falls out of that one expression.

That is the whole argument for Poisson. Not accuracy. Availability of a closed form.

Poisson

Real arrivals

Arrivals on the 1st

Model

Erlang C

closed form

Simulation

or measurement

A number

by Friday

A number

you can defend

Why the real process is not Poisson

Poisson requires that arrivals be memoryless — the probability of an arrival in the next minute does not depend on how long it has been since the last one. Three things break that on the first of the month:

  • Salary timing. A large fraction of the queue arrives within a two-hour window after payroll clears. That is a scheduled burst, not a memoryless process.
  • Self-excitation. A long queue attracts more arrivals, because people see the queue and join it while they are passing. The rate depends on the state.
  • Opening-time clustering. The process is non-stationary across the day in a way that no single λ\lambda can capture.

A process with those three properties is closer to a Hawkes process with a time-varying baseline than to a Poisson process, and a Hawkes process has no Erlang C.

The thing to take away

The interesting question is not “which model is right”. It is: what decision does the closed form let you make that the right model does not, and is that decision worth the error?

If the answer is “how many tellers to staff on the first of the month”, then a simulation answers it too, and answers it correctly. The closed form is not buying you the decision. It is buying you the meeting.