Erlang C calculator

The Erlang C formula, explained

Erlang C answers one question: with this many calls and this many agents, how many callers wait, and for how long? It assumes calls arrive at random, every caller waits until answered, and any free agent takes the next call. The calculator tries one agent count after another until both your targets are met.

1. The workload, in Erlangs. Calls times average handle time, divided by the interval's length. 100 calls in 30 minutes at 240 seconds each is 100 × 240 ÷ 1,800 = 13.3 Erlangs: as if 13.3 people were on the phone the whole half hour. That's the least you could ever get by with, and almost no one would be answered on time.

2. The chance a caller waits. With A Erlangs and N agents (more agents than Erlangs):

X = A^N ÷ N! × N ÷ (N − A)
S = the sum of A^k ÷ k!, for k from 0 to N − 1
P(wait) = X ÷ (S + X)

3. The service level: the share of calls answered within your target time t.

Service level = 1 − P(wait) × e^(−(N − A) × t ÷ AHT)

4. The average speed of answer and occupancy.

Average speed of answer = P(wait) × AHT ÷ (N − A)
Occupancy = A ÷ N

For the 13.3 Erlangs above, 16 agents answer 69% of calls within 20 seconds and 17 answer 81%, so 17 is the fewest that meet 80% in 20 seconds, with each agent busy 78% of the time. The calculator shows 16.9 for these numbers, so 17 whole people (why fractional agents). When your maximum occupancy needs more agents than the service level does, it sets the number instead. The calculator shows both, and the agent counts either side, so you can see what one more or one fewer does.

Chat: several conversations at once

Erlang C assumes an agent takes one contact at a time, which isn't true of chat. An agent with three chats open answers each more slowly, and chats come in bunches. So the calculator works chat out from the work instead:

• The work: chats times average handle time, the whole session from start to close.
• What one agent carries: chats at once times efficiency. Three at once at 85% efficiency is 2.55 chats' worth of work at a time.
• Room for bursts when several chats arrive together, then your maximum occupancy as a cap.

Your service level target is shown as it is, not predicted: this calculator doesn't predict one for chat. Without your own figures, start from 3 chats at once, 85% efficiency and 80% answered within 60 seconds.

Shrinkage: from agents on the phones to agents scheduled

Erlang C gives the agents who need to be taking contacts. Some of the people you pay won't be: they're off sick, on leave, in training, coaching or meetings. That share of paid time is shrinkage, and the people to schedule are:

Agents scheduled = agents on the phones ÷ (1 − shrinkage)

17 agents on the phones with 25% shrinkage is 17 ÷ 0.75 = 22.7, so 23 people. Leave out breaks and lunches already written into each shift: they're off the phones already, and counting them as shrinkage too would count them twice. Your own history of absence, training and meetings is the best guide to the figure.

Service level, and how it's calculated

Service level is the share of contacts answered within a target time, written as 80/20: 80% within 20 seconds. Your phone system measures it after the fact, usually as the calls answered within 20 seconds divided by the calls offered. Some contracts divide by the calls answered instead, or leave out callers who hang up within a few seconds, so check which yours uses.

Before the fact, Erlang C predicts it with the formula above. 80/20 is the common target for voice, and clients' contracts often set their own. Chat is usually measured in seconds to the first reply, and email in hours.