Erlang C, explained

What the model assumes

• Contacts arrive at random, independently of one another, at a steady average rate within the interval.
• Handle times vary at random around their average.
• Every caller waits until answered: nobody hangs up and nobody gets a busy signal.
• Any agent can take any contact, one at a time, and the next caller in line takes the next free agent.

The formulas

Offered load

Contacts per interval × AHT in seconds ÷ (interval minutes × 60)In Erlangs: the agents busy every moment if contacts arrived evenly.As published in the Staffing Requirement Calculator method, version 1.2.

Erlang C

Erlang B by recurrence, B(n) = A × B(n−1) ÷ (n + A × B(n−1)); then C = B ÷ (1 − (A ÷ n) × (1 − B))The probability a contact waits. The recurrence stays finite at any size.As published in the Staffing Requirement Calculator method, version 1.2.

Service level

1 − C × e^(−(agents − load) × threshold seconds ÷ AHT)The share answered within the threshold.As published in the Staffing Requirement Calculator method, version 1.2.

Speed of answer

C × AHT ÷ (agents − load)The average wait across all calls.As published in the Schedule Adherence Impact Calculator method, version 1.1.

A worked example

A load of 10 Erlangs, for example 200 calls an hour at 180 seconds each, with a target of answering within 20 seconds. Each row adds one agent. Computed by the same engine the Schedule Adherence tool runs.

Adding agents brings the wait down fast at first and then slowly, while occupancy falls with every agent added. That trade is the staffing decision.

Checked against

• Erlang C probability of waiting, 10 Erlangs. Offered load 10 Erlangs on 11, 12, 13 and 14 agents. Expected: 0.6821, 0.4494, 0.2853 and 0.1741. Published value; Erlang C formula (A. K. Erlang, 1917), as tabulated in standard Erlang C probability-of-wait tables.
• Erlang C by hand, 1 Erlang on 2 agents. Offered load 1 Erlang on 2 agents. The formula gives (1²/2! × 2/(2 − 1)) ÷ (1 + 1 + 1²/2! × 2/(2 − 1)) = 1 ÷ 3. Expected: 0.3333. Worked by hand from the formula; Erlang C formula (A. K. Erlang, 1917).
• Agents for 80% in 20 seconds at 10 Erlangs. 100 calls a half hour at 180 seconds (10 Erlangs), target 80% answered in 20 seconds. Expected: 14 agents. Published value; Standard Erlang C staffing tables for a 10 Erlang load at an 80/20 service level.

Our test suite computes each case with the shipped engine and fails if any answer differs.

Where it falls short

• Because nobody hangs up in the model, it slightly overstates the agents needed when callers do abandon. The Staffing Calculator shows an abandonment check beside the result.
• It assumes agents take one contact at a time, so it does not fit chat or email, where an agent handles several at once.
• It plans one interval. A day needs the calculation for each interval, and shrinkage on top.

Measure yours

• Erlang C Staffing Calculator (method)
• Schedule Adherence Impact Calculator (method)
• Occupancy Risk Simulator: Occupancy vs Burnout (method)