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Queue Length Estimate Calculator

Work out queue length estimate instantly with clear inputs, formula shown and shareable results.

About Queue Length Estimate Calculator

This uses the M/M/1 result with the servers pooled into a single service rate: utilisation is arrivals over capacity and the average queue is utilisation squared over one minus utilisation. Queue length explodes as utilisation approaches one, which is why a till running at 95% capacity has a queue nineteen times longer than one at 50%.

Formulas

Utilisation

rho = arrival rate / (servers x service rate per server)

Queue length

Lq = rho^2 / (1 - rho)

Worked example

With the default values below, Queue Length Estimate Calculator returns Average people waiting of 3.200. Change any input above and every figure here is recalculated in your browser.

Inputs used

Arrivals per hourminimum 0.1
40
Customers one server handles per hourminimum 0.1
50
Number of serversminimum 1
1

Results

Average people waiting
3.200
Server utilisation
80.00%
Average people in the system
4.000
Queue stability
Stable queue

Step by step

  1. 1Values used
  2. 2Utilisation
  3. 3Queue length
  4. 4Average people waiting= 3.200
  5. 5Server utilisation= 80.00
  6. 6Average people in the system= 4.000
  7. 7Queue stability= Stable queue

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 3 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How is the Queue Length Estimate figure worked out?

rho = arrival rate / (servers x service rate per server). This uses the M/M/1 result with the servers pooled into a single service rate: utilisation is arrivals over capacity and the average queue is utilisation squared over one minus utilisation.

Why does this matter?

Queue length explodes as utilisation approaches one, which is why a till running at 95% capacity has a queue nineteen times longer than one at 50%.

What do I need to enter?

Enter arrivals per hour, customers one server handles per hour, number of servers. The defaults shown are a realistic worked example — swap in your own figures to get a result you can use.

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