Free Queuing Theory Calculator

Queue wait time calculator and staffing planner

Enter arrivals per hour, average service time and the number of counters to get the average wait, queue length, utilisation and the staff you need to hit a target wait. Uses the Erlang C (M/M/c) model, with an hour-by-hour staffing plan and a free PDF report.

Start from an example:

Examples are illustrative starting values. Replace them with your own counts for a real answer.

Results

Average wait in the queue

–

Staff utilisation: –
Chance a customer waits: –
Average queue length: –
Average time in system: –
Wait longer than target: –
90% wait no longer than: –
Customers per day: –
Staff needed (average hour): –
Staff needed (busiest hour): –

Long queue at reception?

With Vizitor pre-registration, visitors enter their details before they arrive and check in with a QR code, so each check-in takes less time at the desk. Lower the service time above to see what a quicker check-in does to your wait.

See visitor pre-registration →

Staffing table

Staff	Utilisation	Chance of waiting	Average wait	Wait > target	90% wait under	Avg queue length	Meets target

One shared line vs a line per counter

Same staff and arrivals. Shared line: everyone joins one queue and goes to the next free counter (M/M/c). Separate lines: each counter has its own queue, customers pick one at random and never switch (M/M/1 per line).

	One shared line	Separate lines

Busiest hour

Assumes random (Poisson) arrivals, exponentially distributed service times, first come first served and no one leaving the queue. How the maths works

How To Use

How to calculate queue wait time in 3 steps

Enter your arrivals and service time

Count how many customers, patients or visitors arrive in a typical hour and time how long one takes at the counter. Add how many staff serve at once, your opening hours and, if you have a rush, a peak-hour multiplier. Or start from an example: clinic, bank, visitor reception, retail or government office.

Set a target wait

Choose the longest wait you are happy with, and whether the average wait must be under it or a share of customers (80%, 90% or 95%) must be served within it. The calculator finds the smallest number of staff that meets your target, for an average hour and for the busiest hour.

Plan staff and share the report

Compare staff levels in the staffing table, see why one shared line beats separate lines, and plan staff hour by hour. Download a PDF staffing report or a CSV of the table, print it, or copy a short summary for your team.

The Formula

How the maths works: the Erlang C (M/M/c) queue model

The calculator uses the M/M/c queue, the standard queuing theory model for a single line served by several identical counters. It is the same Erlang C formula used to staff call centres and service desks. Here is every formula it uses.

1. Workload and utilisation

λ is arrivals per hour, S is average service time in minutes and c is the number of staff. The workload, in staff, is how many people would be busy all the time:

a = λ × S / 60 ρ = a / c (utilisation)

If ρ is 100% or more, staff cannot keep up and the queue grows without limit, so you need more than a staff to have a stable queue.

2. Chance of waiting (Erlang C)

First the Erlang B value is built up one counter at a time, which stays accurate even for hundreds of counters. Erlang C, the chance a new arrival has to wait, follows from it:

B(0) = 1 B(k) = a·B(k−1) / (k + a·B(k−1)) C = B(c) / (1 − ρ·(1 − B(c)))

3. Wait, queue length and time in system

The average wait Wq is across all customers, including those served straight away. Queue length follows from Little's law.

Wq = C × S / (c − a) Lq = (λ / 60) × Wq W = Wq + S

4. Waits longer than the target

The share of customers who wait longer than t minutes, and the wait that 90% of customers stay within:

P(wait > t) = C × e^(−(c − a)·t / S) P90 = S / (c − a) × ln(C / 0.1) (0 if C ≤ 10%)

5. Staff needed

Starting from the fewest staff that can keep up, the calculator adds one at a time until your target is met. It stops at a + 10√a + 50 staff, far beyond the point where waits become negligible, so a result is always found for valid inputs.

6. Assumptions

Arrivals are random (Poisson) at a steady rate, service times are exponentially distributed, customers are served first come first served and nobody gives up and leaves. Real queues with very regular service times usually wait a little less, and a rush that builds a backlog can take longer to clear than the hourly figures suggest.

Reading Your Results

What each queue measure tells you

⏱️

Average wait

The average time a customer spends in the line before service starts. It includes people who walk straight up to a free counter, so many customers wait less and some wait much longer.

📊

Utilisation

The share of time staff are busy serving. Waits rise slowly at first and then very steeply as utilisation approaches 100%, which is why a team that is "only" 90% busy can still have long queues.

🎯

Wait longer than target

The share of customers who wait longer than your target. A service level such as "80% served within 3 minutes" is often a fairer goal than an average, because it limits how many people have a bad wait.

👥

Queue length

The average number of people standing in line. Use it to size your waiting area, seating and floor markings for the busiest hour, not just the average one.

🔀

Shared line vs separate lines

With the same staff, one shared line feeding all counters gives shorter and fairer waits than a separate line at each counter, because no counter sits idle while another has a queue.

📅

Busiest hour and hourly plan

Staffing for the daily average leaves the rush understaffed. Enter a peak-hour multiplier or arrivals for each hour to see where extra staff, or a quieter appointment slot, is needed.

Who It's For

Queue and staffing planning for every front desk

🏥

Clinics and hospitals

Work out how many reception or registration staff you need to keep patient check-in waits short during the morning rush.

🏦

Banks and branches

Decide how many tellers to open by hour, and check how the lunchtime peak changes the wait at the counter.

🏢

Office and visitor reception

See whether one receptionist can handle the morning arrival rush, and how much a faster visitor sign-in would cut the queue.

🛒

Retail checkouts

Plan how many tills to open through the day so most shoppers are served within your target wait.

🏛️

Government and public service offices

Size counters for licence, permit and citizen service desks where service takes longer and demand is high.

🎟️

Events, campuses and service desks

Estimate registration desk staffing for events, student services and IT help desks with a walk-up queue.

Practical Tips

How to reduce queue wait time without guesswork

Measure before you plan

Count arrivals in each hour for a week and time 20 to 30 services. Real numbers beat estimates, and the hour-by-hour plan shows where the problem really is.

Shorten the service, not just the line

Every minute cut from each service reduces the workload for every customer. Collecting details in advance, through pre-registration or online forms, is often cheaper than adding a counter. Try a lower service time in the calculator to see the effect.

Use one line for all counters

A single snake line or a token system that calls the next customer to any free counter is usually faster and feels fairer than separate lines.

Staff for the peak, flex the rest

Move staff from quiet hours into the rush, stagger breaks away from the peak and spread appointments to flatten demand.

Vizitor Queue Management System

How Vizitor reduces queue time

Pre-registration

Visitors, patients and customers fill in their details online before they arrive. Front desk staff skip data entry and go straight to serving, so each service takes less time.

Digital token system

Visitors receive a digital token on arrival and are called to the next free counter. They can sit, move around or wait outside instead of standing in a crowded line.

SMS notifications

Automatic SMS alerts tell people when their turn is approaching, and staff can send delay messages so everyone in the queue stays informed.

Trusted by 500+ workplaces worldwide · 4.8/5 on Capterra · 4.3/5 on G2

Ready to cut your queue wait time?

This calculator tells you how many staff you need. Vizitor shortens the queue itself, with visitor pre-registration, QR check-in, digital tokens and SMS alerts for clinics, banks, offices and public service desks.

Explore Queue Management System See Token System

✅ Free plan available ✅ Setup in under a day ✅ Any tablet ✅ ISO 27001 certified

FAQS

Frequently Asked Questions

How do you calculate queue wait time?

Work out the workload a = arrivals per hour × service time in minutes ÷ 60. With c staff, utilisation is a ÷ c. The Erlang C formula then gives the chance a customer has to wait, C, and the average wait is C × service time ÷ (c − a). This calculator does all of it for you and shows every formula in the “How the maths works” section.

What is the Erlang C formula?

Erlang C gives the probability that an arriving customer has to wait in an M/M/c queue: one line, c identical servers, random arrivals and random service times. It is calculated from the Erlang B value as C = B / (1 − ρ(1 − B)), where ρ is utilisation. It is the standard formula for staffing call centres, service desks and counters.

How many staff do I need to keep the wait under a target?

Enter your arrivals per hour, average service time and target wait, then choose whether the average wait must be under the target or 80%, 90% or 95% of customers must be served within it. The calculator adds staff one at a time from the smallest number that can keep up until the target is met, for an average hour and for your busiest hour.

What does the example in the calculator show?

With 20 arrivals an hour, a 6 minute service time and 3 counters, staff are 67% busy, 44% of customers have to wait, the average wait is about 2.7 minutes, about 0.9 people are queuing on average and 90% of customers wait no longer than about 9 minutes.

Why does the wait grow so fast as staff get busier?

Arrivals are random, so customers bunch up. When staff are busy most of the time there is little spare capacity to clear those bunches, and waits rise steeply as utilisation approaches 100%. At 100% or more the queue never settles and keeps growing, which the calculator flags as at capacity or overloaded.

Is one shared line better than a line per counter?

Yes, with the same staff and arrivals. With one shared line the next customer goes to whichever counter is free, so no counter sits idle while another has a queue. The calculator compares both, treating separate lines as independent single-counter queues where customers pick a line at random and do not switch.

Can I plan staffing for different times of day?

Yes. Add a peak-hour multiplier to check your busiest hour, or tick “Plan staffing hour by hour” and enter the arrivals for each opening hour. You get the staff needed in each hour, the wait with your current staff, total staff-hours for the day and the most staff needed at once.

What assumptions does the calculator make?

It uses the M/M/c model: random (Poisson) arrivals at a steady rate within each hour, exponentially distributed service times, one first-come-first-served line and no customers leaving the queue. Queues with very regular service times usually wait a little less, and a backlog built up in a rush can take longer to clear than steady-state figures suggest.

What do I get in the PDF report?

The PDF staffing report includes your inputs, all results, the staffing table, the shared line vs separate lines comparison, your hour-by-hour plan if you made one, and the formulas used. You can also download the staffing table as a CSV for Excel or Google Sheets, print the report, or copy a short summary.

Is this queue calculator free?

Yes, it is free with no account needed. We ask for your name and work email the first time you download or print a report in a session, so we can send you queue and reception tips. Your inputs are calculated and saved in your own browser.

How can I reduce wait times at reception?

Either add staff at the busiest times or shorten each service. Cutting service time helps every customer, so try a lower service time in the calculator to see the effect. Vizitor’s pre-registration lets visitors enter their details before they arrive and check in with a QR code, which reduces time spent at the desk.