M/M/1 · single-server queue
Queue The line that explodes as ρ → 1
Customers arrive by luck and are served by luck. While the server keeps ahead, the line breathes. Let arrivals creep up on service — utilisation ρ toward one — and the average wait doesn't just rise. It runs to infinity.
Mean wait W = 1 / (1 − ρ) — at ρ = 0.85, that is 6.7 service-times; at ρ = 0.98, fifty.
A server at seventy percent feels fine. At ninety-eight, the same machine feels broken — and the arithmetic is why.
01 Arrivals are Poisson
Gaps between customers are exponential and memoryless: how long you have already waited tells you nothing about the next arrival. Rate λ sets the pace of the door.
02 Service is exponential
One server, one job at a time, each taking a random exponential spell at rate μ. Fast on average, but the occasional long job lets a backlog gather behind it.
03 The blow-up
Mean wait is W = 1 / (μ − λ). As ρ = λ/μ approaches one the denominator vanishes, so the wait — and the line — diverge. Chance becomes a wall.