Queue Wait Time Estimator
Estimate wait time in a queue: position × mean service time. Accounts for parallel servers (counters).
M/M/1 queue: waiting-time model
The M/M/1 model assumes Poisson arrivals (rate λ), exponential service times (rate µ) and one server. The system is stable only when utilisation ρ = λ / µ < 1. Average number in the system L = ρ / (1 − ρ); average time in the system W = 1 / (µ − λ). Example: λ = 40/h, µ = 60/h → ρ = 0.667, L = 2 customers, W = 1/20 h = 3 min. The counter-intuitive part: at ρ = 0.8, W = 1/(0.2·µ); at ρ = 0.9, W = 1/(0.1·µ) — double the wait. As ρ → 1, queues explode, which is why production systems always plan for overcapacity. With k servers (M/M/k) the steady-state solution is the Erlang C formula, the basis for call-centre staffing.
Applications
Used in web and database server sizing, call-centre staffing (Erlang C), data-centre SLA capacity planning, restaurant table turnover, emergency-room triage, traffic-light timing and intersection design. Whenever requests arrive randomly and service time varies, M/M/1 (or M/M/k) gives the first useful estimate.
FAQ
Why does 90% utilisation feel so much worse than 80%? Because waiting time scales with 1/(1−ρ): going from 0.8 to 0.9 doubles the wait, from 0.9 to 0.95 doubles it again.
Is M/M/1 realistic? Real arrivals are usually close to Poisson; service times are rarely truly exponential. M/M/1 gives a useful upper bound and is easy to compute by hand.
How do I size for k servers? Use Erlang C (M/M/k): pick the smallest k such that the probability of waiting and the mean wait meet your SLA.
What if ρ ≥ 1? The queue grows without bound. You must add capacity, throttle arrivals or shed load.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.