# Twinworks Queueing Check

> M/M/1, M/M/c, M/D/1, M/G/1 and Kingman waiting times — optionally verified by simulation. Closed-form queueing results for a single workstation: utilization, mean waiting time, queue length and time in system for M/M/1, M/M/c (Erlang C), M/D/1, M/G/1 (Pollaczek–Khinchine) and the Kingman G/G/1 approximation. Optionally runs the Twinworks simulation of the same station and reports whether the theory lies inside the simulated confidence interval.

Part of usefulweb.site (https://usefulweb.site/). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://twinworksqueueformulas.usefulweb.site/ (interactive tool at https://twinworksqueueformulas.usefulweb.site/tool)
- JSON API: POST https://twinworksqueueformulas.usefulweb.site/api/v1/calculate with a JSON object of inputs (schema: https://twinworksqueueformulas.usefulweb.site/api/v1/tool)
- OpenAPI: https://twinworksqueueformulas.usefulweb.site/openapi.json
- MCP (Streamable HTTP): https://twinworksqueueformulas.usefulweb.site/mcp - tool `twinworks_queue_formulas`

## Inputs
- queue (select): Queue model
- arrivalsPerHour (number, required): Arrival rate
- serviceTime (number, required): Mean service (cycle) time
- servers (integer): Servers (M/M/c)
- serviceScv (number): Service SCV (variance ÷ mean²)
- arrivalScv (number): Inter-arrival SCV (G/G/1)
- verify (boolean): Verify by simulation - Runs the same station in Twinworks (M/M/1, M/M/c, M/D/1 only).
- replications (integer): Replications for the check

## Method
- M/M/1: Wq = ρ/(μ−λ). M/M/c: Wq = C(c, a)/(cμ−λ) with the Erlang C probability C computed by a stable recursion. M/G/1: Wq = λE[S²]/(2(1−ρ)) (Pollaczek–Khinchine), M/D/1 is the special case SCV = 0. Kingman: Wq ≈ ((ca²+cs²)/2)·(ρ/(1−ρ))·te.
- L = λW and Lq = λWq (Little’s law).
- Verification builds a source → unlimited queue → station → sink model, simulates long replications (≈ 20 000 arrivals each, 5 % warm-up) and checks whether the theoretical Wq lies inside the Student-t interval of the simulated mean wait.

## Limits
- Formulas assume steady state, infinite queue capacity and FIFO service. Kingman is an approximation.