Queueing Theory Calculator

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M/M/1 Queueing Theory Calculator

Calculate utilization, queue length, waiting time, and empty-system probability for a single-server M/M/1 queue.

Last updated: June 2026 | By Patchworkr Team

Calculation steps
rho = lambda / mu = 10 / 15 = 0.666667
L = lambda / (mu - lambda) = 2
Lq = lambda^2 / (mu(mu - lambda)) = 1.333333
W = 1 / (mu - lambda) = 0.2
Wq = lambda / (mu(mu - lambda)) = 0.133333
Queue metrics
rho = 0.666667, L = 2, Lq = 1.333333
Utilization rho
66.666667%
Empty probability P0
33.333333%
Average number in system L
2
Average number in queue Lq
1.333333
Average time in system W
0.2
Average time in queue Wq
0.133333

What Is M/M/1 Queueing Theory?

An M/M/1 queue models a single server system where arrivals follow a Poisson process and service times are exponentially distributed.

The calculator uses the standard steady-state formulas for utilization, average queue length, average system length, waiting time, and the probability that the system is empty.

How To Use The Queueing Theory Calculator

  1. Enter the arrival rate lambda.
  2. Enter the service rate mu.
  3. Make sure lambda is less than mu so the queue stays stable.
  4. Read the live utilization and waiting-time metrics in the result panel.
rho = lambda / mu

Worked Example

If lambda = 10 and mu = 15, then rho = 0.666667 and the average number in system is 2.

rho = 10 / 15 = 0.666667
L = 10 / (15 - 10) = 2
W = 1 / 5 = 0.2

Frequently Asked Questions

Why must lambda be smaller than mu?

If arrivals are not slower than service, the queue grows without bound and the steady-state formulas do not apply.

What does rho represent?

Rho is the utilization of the single server, or the fraction of time the server is busy.

Can I use decimal rates?

Yes. The calculator accepts any positive finite numeric rates.

Does this support multi-server queues?

No. This tool is for the standard M/M/1 case with one server.

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