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Erlang A: staffing when customers hang up

What Erlang A adds to Erlang C, how average patience changes the answer, and why impatient callers cost abandonment while patient ones cost agents.

Published

Erlang C has one assumption that everyone knows is false: nobody ever hangs up. It is a safe assumption when staffing is generous, because few people wait long enough to consider it. It fails exactly when you need the model most, at the tight end, where Erlang C predicts queues that in reality turn into abandoned calls. Erlang A fixes that by giving every waiting customer a patience.

What changes

Erlang A adds one input, average patience: the mean time a customer will wait before giving up. With it, the model predicts three things Erlang C cannot:

  • Abandonment rate, the share of contacts that hang up before being answered.
  • A service level that counts abandons. Answered within the target divided by offered, which is what an ACD reports.
  • The wait of the answered. Average speed of answer over the customers who stayed, which is lower than Erlang C’s figure because the longest waiters have left the queue.

The queue also becomes stable with fewer agents than erlangs of work, because departures no longer depend on agents alone. That is the mathematical reason Erlang C gives a zero service level at 34 agents for 33.3 erlangs while a real centre with 34 agents limps through the interval losing a tenth of its callers.

Worked example: same interval, four levels of patience

Take the reference interval used across these guides: 200 contacts in 30 minutes, a 300-second handle time, 33.3 erlangs, an 80/20 target, and a ceiling of 5 per cent abandonment.

Average patienceAgentsService levelAbandonedErlang C would say
60 s3791.3%4.7%39
90 s3789.5%4.2%39
180 s3682.1%4.4%39
600 s3883.6%1.5%39

Two things stand out. First, in every case Erlang A needs fewer agents than Erlang C’s 39, because the customers who leave are not in the queue. Second, the pattern is not monotone. Very impatient customers (60 seconds) are lost quickly, so the queue stays short and service level is high, but abandonment is what binds: 36 agents would lose more than 5 per cent. Patient customers (600 seconds) wait rather than leave, so abandonment is low but the queue is long, and service level is what binds. Impatient callers cost you abandonment; patient callers cost you agents.

What tight staffing actually looks like

The other use of Erlang A is describing an interval you know is short, which Erlang C cannot do sensibly. With 90-second patience:

AgentsErlang A service levelAbandonedErlang A ASAErlang C service levelErlang C ASA
3480.5%7.9%6 s16.9%391 s
3686.9%5.3%4 s53.8%62 s
3891.8%3.3%3 s75.6%21 s
3993.7%2.6%2 s82.6%13 s

At 34 agents Erlang C predicts an average wait of six and a half minutes and a service level of 17 per cent. Erlang A predicts an 80 per cent service level and an ASA of six seconds, with 8 per cent of callers gone. Both are describing the same interval. The Erlang C numbers are what the queue would look like if nobody left; the Erlang A numbers are what your ACD will show, and they explain why a visibly short-staffed interval can still post a respectable service level.

How to use it

  • Plan with Erlang C, explain with Erlang A. The requirement should not have an abandonment rate baked into it unless the business has agreed to lose those customers. Erlang A is for understanding the intervals where you will be short anyway.
  • Estimate patience from time to abandon, carefully. The ACD’s average time to abandon only sees customers whose patience ran out, so it understates true patience when service is good. Use it as a starting point and calibrate against observed abandonment.
  • Set the abandonment ceiling deliberately. It is often the binding constraint, as the first table shows. A 5 per cent ceiling and an 80/20 target are not the same requirement and it is worth knowing which one you are staffing to.

Try it with your own numbers

Inputs
Demand

Calls, chats or tickets arriving during one interval.

s

Talk plus hold plus wrap-up.

s

Mean time customers wait before hanging up. Estimate it from your average time to abandon.

Targets
%

Share of contacts to answer within the target time.

s
%

Agents are added until the abandon rate is at or below this.

Results
Agents required
37

Meets both the service level and the abandonment ceiling, before shrinkage.

Service level
89.5%

Answered within target ÷ offered.

Abandoned
4.2%
Average speed of answer
3.3 s

Answered contacts only.

Contacts that will wait
21.4%
Occupancy
86.3%
Erlang C would need
39

Same target, no abandonment.

Workload
33.3 erlangs
Show the working
  1. Workload = 200 contacts ÷ 30 min × 300 s ÷ 60 = 33.3 erlangs.
  2. Erlang A is the Erlang C queue with every waiting customer hanging up at a rate of 1 ÷ 90 s. Because customers leave, the queue is stable even with fewer agents than erlangs.
  3. With 37 agents: 21.4% of contacts wait at all, 4.2% abandon, and 89.5% are answered within 20 s. That clears the 80% service level and the 5% abandonment ceiling.
  4. Carried load is 33.3 × (1 − 0.042) = 31.9 erlangs, so occupancy is 86.3%.
  5. Erlang C alone, which assumes infinite patience, would need 39 agents for the same service level.

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.

Where next

Frequently asked questions

Can I use abandon rate as my only target?
You can, and some centres do, but it is a weak target on its own: a short queue that everyone abandons quickly and a long queue that a few people give up on can have the same abandon rate. Pair it with a service level so that both the wait and the loss are controlled.
Does Erlang A model retries?
No. A customer who abandons and calls back is a new arrival. If retries are a large share of your volume, the volume you are forecasting already includes them, and reducing abandonment will reduce volume, which Erlang A cannot see. Erlang X extends the model to retrials, but it needs a retry probability that few centres can measure.

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