How to calculate Erlang C staffing for a contact centre

Erlang C is the formula behind almost every staffing number in a contact centre. It answers one question: given how many contacts arrive, how long each one takes, and how quickly you want them answered, how many agents must be available at once? This guide walks through the calculation with a worked example you can follow in the calculator below.

What you need before you start

Three inputs, all for a single planning interval:

• Contacts in the interval. From your forecast. If you plan in 30-minute intervals, this is the volume for one 30-minute slot, not the hourly or daily total.
• Average handle time (AHT) in seconds: talk time plus hold plus after-call wrap-up.
• A service level target. Usually written as two numbers, such as 80/20: 80 per cent of contacts answered within 20 seconds.

The worked example uses 200 contacts in a 30-minute interval, an AHT of 300 seconds, and an 80/20 target.

Step 1: turn volume and handle time into erlangs

An erlang is one hour of work in one hour, or, more usefully, the average number of contacts in progress at any moment. It is arrival rate multiplied by handle time in the same unit:

erlangs = (contacts ÷ interval minutes) × (AHT seconds ÷ 60)
        = (200 ÷ 30) × (300 ÷ 60)
        = 6.67 contacts per minute × 5 minutes
        = 33.3 erlangs

On average, 33.3 conversations are happening at once. That immediately tells you 33 agents is not enough: with as many agents as erlangs the queue never clears, because arrivals are random and bunch up. The question is how many more than 33.3 you need, and that depends on the target.

Step 2: the probability that a contact has to wait

For a given number of agents N and workload A, Erlang C gives the probability that a new contact finds every agent busy:

P(wait) = [ (A^N ÷ N!) × N ÷ (N − A) ]
          ÷ [ Σ (A^k ÷ k!) for k = 0 … N−1  +  (A^N ÷ N!) × N ÷ (N − A) ]

You will not want to do this by hand, and you should not do it with the factorials in a spreadsheet either, because they overflow past a hundred or so agents. The standard approach is a recurrence that walks up from one agent to N. With 39 agents and 33.3 erlangs, the probability of waiting is 25.4 per cent.

Step 3: from probability of waiting to service level

Contacts that do not wait are answered within the target by definition. For the ones that do wait, the chance of still being in the queue after T seconds falls away exponentially:

service level = 1 − P(wait) × e^( −(N − A) × T ÷ AHT )
              = 1 − 0.254 × e^( −(39 − 33.3) × 20 ÷ 300 )
              = 82.6%

The exponent is where the intuition lives. N − A is spare capacity: the more agents you have above the workload, the faster the queue drains. T ÷ AHT is how generous the target is relative to handle time: a 20-second target against a 5-minute AHT is demanding; the same target against a 90-second AHT is much easier.

Step 4: search for the smallest number of agents

Erlang C does not solve for N directly. You try agent counts until the service level clears the target:

Agents	Contacts that wait	Service level	ASA	Occupancy
36	55.2%	53.8%	62 s	92.6%
37	43.2%	66.2%	35 s	90.1%
38	33.3%	75.6%	21 s	87.7%
39	25.4%	82.6%	13 s	85.5%
40	19.1%	87.8%	9 s	83.3%

The answer is 39. Notice how steep the curve is near the target: each agent moves service level by around ten points. Being one agent short does not cost you a little service level, it costs a lot, which is why intervals with a single late log-in can fail.

Step 5: check occupancy, then add shrinkage

Two things remain before the number is usable.

Occupancy. 33.3 erlangs across 39 agents is 85.5 per cent occupancy: agents are handling contacts for 85.5 per cent of their available time. That is on the edge of what people can sustain for a full shift. Many planners cap occupancy at 85 per cent, which would round this interval up to 40 agents. The occupancy guide covers why.

Shrinkage. 39 is the number of agents who must be logged in and available for the whole interval. It is not the number to put on the roster, because some of the people you schedule will be on a break, in a coaching session, or off sick. At 30 per cent shrinkage you need 39 ÷ 0.7 = 55.7, so 56 scheduled. The shrinkage guide shows how to get your own figure, and the staffing requirement calculator runs all three steps together.

Try it with your own numbers

Inputs

Reset

Demand

Contacts in the interval

Calls, chats or tickets arriving during one interval.

Interval length15 minutes30 minutes60 minutes

Average handle time

s

Talk plus hold plus wrap-up.

Targets

Service level target

%

Share of contacts to answer within the target time.

Target answer time

s

Results

Copy results

Agents required

39

On the phones for the whole interval, before shrinkage.

Service level achieved

82.6%

Occupancy

85.5%

Average speed of answer

13.4 s

Contacts that will wait

25.4%

Workload

33.3 erlangs

Show the working

1. Workload = 200 contacts ÷ 30 min × 300 s ÷ 60 = 33.3 erlangs.
2. The queue is only stable with more agents than erlangs, so the search starts at 34.
3. With 39 agents, Erlang C gives a 25.4% chance that a new contact finds every agent busy.
4. Service level = 1 − 0.254 × e^(−(39 − 33.3) × 20 ÷ 300) = 82.6%, which clears the 80% target.
5. With 38 agents it would be 75.6%, which does not.
6. Occupancy = 33.3 ÷ 39 = 85.5%.

Doing this for every interval of the week? Pebble WFM computes the requirement from your forecast and builds the roster. Free month, no card needed.

Common mistakes

• Mixing units. AHT in minutes with a target in seconds, or hourly volume in a 15-minute interval, gives an answer that is wrong by a factor of four or sixty and looks plausible.
• Using the average interval. Staffing to the day’s average understaffs the peak and overstaffs the trough. Run the calculation for every interval and staff to the curve.
• Applying it to email and back-office work. Erlang C assumes contacts must be handled the moment an agent is free. Work that can wait hours is a workload calculation, not a queueing one.
• Forgetting that it ignores abandonment. Erlang C assumes infinite patience. At very low staffing it predicts queues that would never happen because customers hang up. Treat the model’s numbers near the unstable end with caution.
• Treating the output as headcount. It is agents on the phones for one interval. Headcount comes from covering every interval with shifts, then adding shrinkage.

Where next

• How to calculate service level and ASA: the same maths run the other way, from a staffing level to the service level it will deliver.
• How to calculate a staffing requirement: from this interval’s answer to a week’s roster and an FTE figure.