The two formulas behind every queue – Little’s Law and Erlang C – explained without the maths jargon, with worked examples for sites and medical centres.

Queuing theory sounds academic, but its core ideas are effortless and extremely helpful for anyone who plans personnel for a branch, a hospital department or a public service hall. Two tools do most of the work: Little’s Law and the Erlang C model.

The three numbers that describe any queue

• Arrival rate – how many service users arrive per hour.
• Service time – how long one visitor spends at the counter on average.
• Number of counters – how many personnel are serving simultaneously.

Everything else – wait time, queue length, utilisation – follows from these three. A queue management system records all of them automatically, ticket by ticket.

Little’s Law: individuals in the room = arrivals × time spent

Little’s Law says that the average number of individuals in a system equals the arrival rate multiplied by the average time each person spends in it. If 60 service users arrive per hour and each spends 20 minutes (one third of an hour) waiting, you will have approximately 20 individuals waiting at any moment.

It works in reverse too. If your waiting lounge seats 30 and 90 service users arrive per hour, the average wait must stay under 20 minutes or individuals will be standing. It’s a rapid way to size a waiting hall or to check whether a target wait time is realistic.

Utilisation: the number that explains sudden queues

Utilisation is the share of time counters are crowded. It equals arrivals × service time ÷ counters. With 50 visitors per hour, 6-minute service and 6 counters, utilisation is 50 × 0.1 ÷ 6 ≈ 83%.

The main insight of queuing theory is that wait time doesn’t grow in a straight line with utilisation. Going from 70% to 80% adds a little waiting; going from 90% to 95% can double it. That is why a branch that copes perfectly on a normal day collapses when one member of team is on a break.

Erlang C: predicting the wait

The Erlang C model, first used to plan telephone exchanges, estimates the probability that a visitor will have to wait at all, and how long the average wait will be, for a given number of counters. Call centres still use it every day, and it fits walk-in service halls well when service users arrive at random.

You don’t have to do the maths by hand. Our free wait-time calculator uses Erlang C: enter arrivals per hour, service time and counters, and it presents the expected wait, service level and how many counters you have to hit a target.

A worked example

A bank branch sees 72 clients per hour at lunchtime, with an average service of 5 minutes. That is 6 hours of work arriving every hour, so at least 7 counters are needed just to keep up. With 7 counters, utilisation is about 86% and the average wait is a few minutes; with 6 counters the queue never stops growing. One extra counter at the peak is the difference between a calm lunchtime and a lost afternoon.

Where theory meets your real data

• Use real arrival curves by hour and weekday from your queue reports, not daily averages.
• Track service time per service type – a document check and an account opening aren’t the same.
• Re-check after every change: new services, appointments or mobile ticketing all change the arrival pattern.

Combined with live data from QVista, these two formulas let you personnel for the service users you actually have, rather than for the ones you hope for. See also our queue glossary for the terms used here.