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M/M/1 Queue Calculator

Enter the arrival rate λ and service rate μ (λ < μ) to get the M/M/1 queue metrics.

How to use

  1. Enter your values in the fields above.
  2. Press Calculate to see your result instantly.
  3. Use the Share button to copy a link to your result.

About this calculator

The M/M/1 queue is the foundational model of queueing theory: a single server, Markovian (Poisson) arrivals at rate λ, and Markovian (exponential) service times at rate μ — the "M/M/1" name comes directly from Kendall's notation for this arrival/service/server-count combination. This calculator takes λ and μ (with λ < μ required for a stable, non-exploding queue) and returns the standard performance metrics.

It computes utilization ρ = λ/μ, the average number of customers in the system L = ρ/(1−ρ) and in the queue Lq = ρ²/(1−ρ), and the average time in the system W = 1/(μ−λ) and waiting time Wq = ρ/(μ−λ) — via Little's Law, L = λW. Operations researchers, call-center planners and computer-network engineers use these formulas to predict wait times and congestion from nothing more than an arrival rate and a service rate.

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