Why does a line that is 95% busy feel infinitely long? Queueing theory explains it — and the M/M/1 queue is where it all starts.
M/M/1 QueueLive
queueing theory
Controls
Presets
The M/M/1 queue models a single server with random arrivals and service — a checkout, help desk, or router. The utilization ρ = λ/μ decides everything: as it approaches 1, the average wait and queue length explode toward infinity. This nonlinear blow-up is why systems run at, say, 80% and not 99% capacity.
Reading this result: Moderate load: ρ=0.70. On average L≈2.3 in the system waiting W≈3.3. You are on the steep part of the curve — pushing ρ toward 1 makes waits climb far faster than the extra traffic would suggest.
Runs locally in your browser — free forever. Scale to the cloud when reality gets heavy.
★ Sign in to save this setupSave your tuned setup, or drop this simulation into your own site, docs, or course page.
How it works
The M/M/1 queue models a single server with random (Poisson) arrivals and exponential service times. Its utilization ρ = λ/μ governs everything: average queue length and waiting time both blow up as ρ approaches 1. This nonlinear explosion is why call centers, networks, and checkouts are engineered to run below full capacity.
✦
Ask the AI about this model
The math, the assumptions, real-world uses, or a code translation — explained for this exact simulation.
Yes. M/M/1 Queue runs entirely in your browser using your device's own compute, so local use is free forever. You only pay Compute Tokens if you scale a job to the cloud.
Do I need to install anything?▾
No. Everything runs client-side in a modern browser — no downloads, no license, no account required to start.
Can I save or share my simulation?▾
Create a free account to save projects, and use a shareable embed or minted DOI to publish a live, interactive version anywhere.
How accurate are the results?▾
The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.