Queue Wait Time Calculator

Work out queue wait time instantly with clear inputs, formula shown and shareable results.

About Queue Wait Time Calculator

For a multi-server queue, Erlang C gives the probability of waiting and the average wait is that probability times service time divided by the spare server capacity. Queue length then follows from Little's law. The behaviour to notice is that wait time depends on the gap between servers and offered traffic, so the same utilisation is far more painful with few servers than with many.

Formula

M/M/c waiting

traffic = arrivals x service time / 3600; average wait = C x service time / (servers - traffic); queue length = wait x arrival rate

Worked example

With the default values below, Queue Wait Time Calculator returns Average wait in the queue of 24.55. Change any input above and every figure here is recalculated in your browser.

Inputs used

Arrivals per hourminimum 0.01: 240
Service time per itemminimum 0.01 s: 120 s
Serversaccepts 1–300: 10

Results

Average wait in the queue: 24.55 s
Probability of having to wait: 40.92%
Average items waiting: 1.64
Server utilisation: 80.00%

Step by step

1Values used
2M/M/c waiting
3Average wait in the queue= 24.55 s
4Probability of having to wait= 40.92
5Average items waiting= 1.64
6Server utilisation= 80.00

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 2 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

Why does a bigger queue with the same utilisation wait less?

Economies of scale in queueing. Twenty servers at 90 percent utilisation have twice the absolute spare capacity of ten at 90 percent, so bursts are absorbed faster.

Does the model assume anything about arrivals?

Yes, Poisson arrivals and exponential service times. Bursty arrivals produce longer waits than the formula predicts.