How this calculator works
The M/M/s model assumes independent Poisson arrivals, exponential service times, identical parallel servers, one shared first-come queue, unlimited waiting space and no abandonment. Erlang C gives the probability an arrival waits. Little's law then connects average queue lengths to average times. Rates are entered per hour; time outputs can switch units. State n includes both waiting customers and those being served.
Formula
a = λ/μ, ρ = a/s < 1. P₀ = [Σ(k=0..s-1) a^k/k! + a^s/(s!(1-ρ))]^-1. C = P₀a^s/(s!(1-ρ)); Wq = C/(sμ-λ); W = Wq + 1/μ; Lq = λWq; L = λW.
Worked example
With two arrivals per hour and one server handling three per hour, utilisation is 2/3. The mean time in the system is 1 hour and mean waiting time is 2/3 hour. The mean queue has 4/3 customers.
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Frequently asked questions
Why reject a full-capacity arrival rate?
At or above capacity the unlimited-queue model has no finite steady-state mean. Real queues may abandon or reject arrivals, which needs another model.
Is μ the total service rate?
No. It is the rate of each server. Total capacity is s times μ.
What does zero arrivals mean?
The system is empty with probability one. Waiting is zero, while an isolated customer would still require mean service time 1/μ.