Queue Wait Time (M/M/1) Calculator
Model a single-server queue with the M/M/1 formulas: utilisation, average wait time and queue length from arrival and service rates.
How it works
ρ = λ ÷ μ · W = 1 ÷ (μ − λ) · Wq = ρ ÷ (μ − λ) · L = ρ ÷ (1 − ρ) · Lq = ρ² ÷ (1 − ρ) (requires ρ < 1)
The M/M/1 model describes a single server with random (Poisson) arrivals and exponential service times. Its defining feature is that wait time explodes non-linearly as utilisation approaches 100%: the (1 − ρ) in the denominator means going from 80% to 90% busy roughly doubles the queue, and 95% is punishing. That is why capacity planning targets moderate utilisation rather than squeezing every last percent out of a server.
Worked example
With λ = 8/min arriving and μ = 10/min served, ρ = 0.8. Time in system W = 1 ÷ (10 − 8) = 0.5 min (30 s); queue wait Wq = 0.8 ÷ 2 = 0.4 min (24 s); average number in system L = 0.8 ÷ 0.2 = 4.
Sources: Wikipedia — M/M/1 queue
Frequently asked questions
Why does the calculator require ρ < 1?
When arrivals meet or exceed the service rate (ρ ≥ 1), a single server cannot keep up and the queue grows without limit — the steady-state formulas divide by (1 − ρ), which is zero or negative. You need more capacity or fewer arrivals.
What do M/M/1 and the letters mean?
The notation is Kendall’s: the first M is Markovian (Poisson) arrivals, the second M is exponential service times, and the 1 is a single server. It is the simplest useful queueing model.
When is M/M/1 a poor fit?
When you have multiple servers (use M/M/c), when service times are near-constant rather than random (M/D/1 waits are about half), or when arrivals are bursty and not Poisson. Treat M/M/1 as a first-order estimate, not gospel.
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This calculator is for educational purposes. Double-check important results before acting on them.