A single worker, jobs arriving at random. When work comes in slower than it goes out, the line stays short. But push the load toward 100% and the wait doesn’t just rise — it EXPLODES to infinity. The last few percent of capacity cost more than all the rest combined. Slide the load and find the wall.
For a single server with random (Poisson) arrivals and random (exponential) service — the M/M/1 queue — the utilisation is ρ = λ/μ, and the averages are L = ρ/(1−ρ) in the system and W = 1/(μ−λ) of waiting. Both have (1−ρ) in the denominator, so as ρ→1 they diverge to infinity: a system run at 99% load waits ~100× longer than at 50%. A fail-loud self-check throws unless L and W match those formulas and blow up as the load approaches 1 — why you never plan a queue for full utilisation.
M/M/1 assumes Poisson arrivals and exponential service (memoryless); real workloads are burstier or smoother, changing the constant but NOT the (1−ρ) blow-up. The formulas are exact for this classic model.