Erlang C

354. Erlang C

The Erlang C formula: probability that an arriving customer has to wait in an M/M/c queue (Poisson arrivals, exponential service, $𝑐$ servers, infinite buffer).

The exact analog of M/M/c queue’s “no-empty-server probability” at steady state.

354.1. Formula

𝐶 (𝑐, 𝑎) = \frac{\frac{{(𝑐 𝑎)}^{𝑐}}{𝑐!}}{\frac{{(𝑐 𝑎)}^{𝑐}}{𝑐!} + (𝑐 - 𝑐 𝑎) \sum_{𝑘 = 0}^{𝑐 - 1} \frac{{(𝑐 𝑎)}^{𝑘}}{𝑘!}}

Wait — common convention uses offered load $𝑎 = \frac{𝜆}{𝜇}$ (in Erlangs) and number of servers $𝑐$ , with utilization $𝜌 = \frac{𝑎}{𝑐} < 1$ :

𝐶 (𝑐, 𝑎) = \frac{\frac{𝑎^{𝑐}}{𝑐!}}{(1 - 𝜌) \sum_{𝑘 = 0}^{𝑐 - 1} \frac{𝑎^{𝑘}}{𝑘!} + \frac{𝑎^{𝑐}}{𝑐!}}

$𝐶 (𝑐, 𝑎)$ is the probability of waiting (proportion who don’t get served immediately).

354.2. Expected wait

Given $𝐶 (𝑐, 𝑎)$ is the wait probability, the expected wait in queue:

𝐸 [𝑊_{𝑞}] = \frac{𝐶 (𝑐, 𝑎)}{𝑐 𝜇 - 𝜆}

Same form as M/M/1 expected wait, just multiplied by the probability of waiting (since non-waiters contribute zero).

354.3. Where it shows up

Call center staffing: how many agents to keep wait probability below threshold
Bank tellers / DMV staffing
Help desk / IT support sizing
Server farm capacity

354.4. Worked example

Help desk: 30 calls per hour, average call lasts 5 minutes = $\frac{1}{12}$ hr. Offered load: $𝑎 = 30 \cdot \frac{1}{12} = 2.5$ Erlangs.

$𝑐$ (agents)	$𝜌 = \frac{𝑎}{𝑐}$	$𝐶 (𝑐, 𝑎) \approx$
3	0.83	60%
4	0.625	27%
5	0.5	13%
6	0.42	6%

To keep wait probability under 10%, need $𝑐 = 6$ agents. Wait time at $𝑐 = 6$ : $𝐸 [𝑊_{𝑞}] = \frac{0.06}{6 \cdot 12 - 30} = \frac{0.06}{42} \approx 86$ seconds.

354.5. Erlang C in call-center practice

The industry standard formula. Variants:

Square-root staffing rule (Halfin-Whitt regime): $𝑐 \approx 𝜌 + 𝛽 \sqrt{𝜌}$ for some quality-of-service constant $𝛽$
Erlang A: add abandonment — customers leave after waiting too long; see Erlang A
Skill-based routing: agents are heterogeneous; needs simulation

354.6. Limitations

Exponential service times — real call durations are heavier-tailed (often lognormal)
Poisson arrivals — works for unscheduled arrivals; not for scheduled appointments
Infinite buffer — real callers abandon; use Erlang A

For most call-center sizing, Erlang C is good enough. For accuracy use simulation.

354.7. See also

Erlang B — without queue (blocking)
M/M/c — the queue model itself
Erlang A — with abandonment
Square-Root Staffing