Do I need to install anything?

No. It runs in any modern browser, free, with no account required.

For Engineers · M/M/1 Queue

M/M/1 Queue for an ad budget

Built for engineers using it for real design work. Go from concept to a running model in the browser, then scale to the cloud when needed. Simulate an ad budget live below — adjust the inputs and watch it respond, right in your browser.

M/M/1 QueueLive

queueing theory

Controls

Speed1×

Presets

Arrival rate λ

Service rate μ

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.

▶ Run in Python

Data Inspector

Utilization ρ0.70

Avg in system L2.33

Avg in queue Lq1.63

Avg wait W3.33

Live count0

Governing equation

ρ = \frac{λ}{μ} = \frac{0.7}{1} = 0.70, L = \frac{ρ}{1 - ρ} = 2.33, W = \frac{1}{μ - λ} = 3.33

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.

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Frequently asked questions

Is this good for engineers?: Yes — this version of "M/M/1 Queue for an ad budget" is framed for engineers using it for real design work. Go from concept to a running model in the browser, then scale to the cloud when needed.
Do I need to install anything?: No. It runs in any modern browser, free, with no account required.