Steady State M/M/c Queue Workbench

How to use Steady State M/M/c Queue Workbench

Birth-death state weights are built recursively through the server count. A geometric tail normalizes the remaining states. Erlang C gives waiting probability, and Little’s law links expected queue/system counts with waiting/service times.

1. Enter arrival rate per common time unit, service rate per busy server per same time unit, identical servers.
2. Choose the stated convention and check the units and supported model.
3. Calculate, inspect the result, then copy or download it.

Good to know

• Exact-grid inputs are plain decimals with at most six places and magnitude≤1e9. Raw JSON number spelling is checked before conversion; decimal strings can preserve exact grid values. Exact rational intermediates are bounded to 8192 bits. 1–50 identical servers; arrival and per-server service rates share one time unit. Arrival rate must be strictly below servers × service rate. Assumes stationary Poisson arrivals, independent exponential service, one infinite FCFS queue, no abandonment and no finite population. Zero arrivals gives zero queueing delay. Exact-rational budget refusal is possible for extreme parameter combinations. No staffing recommendation, service guarantee or real system control.
• Local calculation only; no input persistence or external lookup. Text/JSON is limited to 30000 characters and output to 1000000 characters. Numeric model inputs and precision/work limits are stated on this page. Approximate JSON numbers retain full stored values; exact arithmetic outputs retain decimal or fraction strings. Unsupported states fail explicitly.
• Educational user-assumption arithmetic only. No live prices, laws, tax rules, quotes or financial advice.