Industrial & Systems
Queue Lab
Queueing theory explains why a checkout, a help desk or a server backs up — and the surprising part is how suddenly. Set the arrival rate λ and the service rate μ; a live stochastic M/M/1 simulation animates customers lining up at one server while the steady-state metrics (utilisation ρ = λ/μ, average number in system L, average wait Wq) show the theory. The non-linearity is the lesson: a server that's 50% busy barely has a line, but at 90% busy the average queue is already nine deep, and at ρ ≥ 1 arrivals outrun the server and the line grows without bound. It's why capacity planning lives or dies on the last 10% of utilisation.
60%
Utilisation ρ
1.50
Avg in system L
0.30
Avg wait Wq
Customers arrive at rate λ and one server clears them at rate μ. The whole system is governed by the utilisation ρ = λ/μ: keep it under 1 and the line stays finite, but as ρ creeps toward 1 the average queue and wait shoot up non-linearly — a 90%-busy server already backs up badly. At ρ ≥ 1 arrivals outrun the server and the queue grows forever.
How to use this simulation
Queueing theory explains why a checkout, a help desk or a server backs up — and the surprising part is how suddenly. Set the arrival rate λ and the service rate μ; a live stochastic M/M/1 simulation animates customers lining up at one server while the steady-state metrics (utilisation ρ = λ/μ, average number in system L, average wait Wq) show the theory. The non-linearity is the lesson: a server that's 50% busy barely has a line, but at 90% busy the average queue is already nine deep, and at ρ ≥ 1 arrivals outrun the server and the line grows without bound. It's why capacity planning lives or dies on the last 10% of utilisation.
Everything runs in your browser — no sign-up, no download. Change a value and the result updates instantly, so you can build a feel for how each input shapes the outcome. It pairs with Crameleon's practice exams and step sheets when you want to go from intuition to working the problems.