Queue Waiting Time Calculator

What this generator does

Computes utilisation, the probability an arrival has to wait at all, and the mean number and time both in the queue and in the system, using the standard M/M/c formulas with Erlang C.

How to use this tool

1. Enter how much work arrives per hour.
2. Enter how fast one server clears work.
3. Set how many servers run in parallel.
4. Read the utilisation first — everything else follows from it.

Understanding the controls

Arrivals per hour

How much work turns up on average. The formulas assume arrivals are random rather than scheduled, which is the realistic case.

Served per hour, per server

How fast a single server clears work when it has some. Multiply by the server count for total capacity.

How many servers

Between 1 and 12 working in parallel from one shared queue — which is why a single queue with several servers beats several separate queues.

Common use cases

• Deciding how many staff, tills or lanes a service needs
• Showing why the last few percent of utilisation cost so much
• Capacity planning for a support desk or a call centre
• Teaching Little's law with numbers that satisfy it
• Checking a simulation's output against the closed form

How this generator works

Erlang C gives the probability that every server is busy when someone arrives; the queue length follows from it, and the waiting time follows from the queue length by Little's law. The check runs Little's law in both directions — for the system and for the queue alone — and requires time in the system to be exactly the wait plus one service time.

Randomness and fairness

Nothing here is random. The same rates always give the same answer, which is why this is a calculator rather than a generator and has no seed.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

• Assumes random arrivals and random service times — the M/M/c model. Scheduled arrivals or fixed service times give shorter waits than these figures.
• One shared queue feeding all servers. Separate queues per server are worse, and this does not model them.
• No limit on queue length, no customers giving up and leaving, and no priority classes.
• Averages only: half of all waits will be longer than the mean, and the tail matters more than the mean for service targets.
• Nothing is stored between visits, so note the figures before changing the rates.

Privacy and your data

Every figure is computed in your browser from the numbers you type. Nothing you enter is transmitted or stored.