M/M/c (Erlang C) Queue Calculator

Poisson arrivals (λ), exponential service (μ), c identical servers. Client-side only.

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Inputs

Uses Erlang C for P(wait). Assumes infinite buffer and FCFS.

Outputs

Utilization ρ = λ/(cμ)
Stability requires ρ < 1
P(wait) (Erlang C)
Prob an arrival must queue
Lq (mean in queue)
Wq (mean wait time)
W (mean time in system)
W = Wq + 1/μ
L (mean in system)
L = λW (Little's Law)
Formulas (summary)
Let a = λ/μ (offered load) and ρ = a/c.
  • P0 = [ Σ_{n=0}^{c-1} a^n/n! + (a^c/c!) * (1/(1-ρ)) ]^{-1}
  • P(wait) = (a^c/c!) * (1/(1-ρ)) * P0
  • Lq = P(wait) * ρ / (1-ρ)
  • Wq = Lq / λ
  • W = Wq + 1/μ
Practical note
This returns means and P(wait). Tail latency needs more modeling.