351. Erlang C
The Erlang C formula: probability that an arriving customer has to wait in an M/M/c queue (Poisson arrivals, exponential service, servers, infinite buffer).
The exact analog of M/M/c queue’s “no-empty-server probability” at steady state.
351.1. Formula
Wait — common convention uses offered load (in Erlangs) and number of servers , with utilization :
is the probability of waiting (proportion who don’t get served immediately).
351.2. Expected wait
Given is the wait probability, the expected wait in queue:
Same form as M/M/1 expected wait, just multiplied by the probability of waiting (since non-waiters contribute zero).
351.3. 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
351.4. Worked example
Help desk: 30 calls per hour, average call lasts 5 minutes = hr. Offered load: Erlangs.
| (agents) | ||
|---|---|---|
| 3 | 0.83 | 60% |
| 4 | 0.625 | 27% |
| 5 | 0.5 | 13% |
| 6 | 0.42 | 6% |
To keep wait probability under 10%, need agents. Wait time at : seconds.
351.5. Erlang C in call-center practice
The industry standard formula. Variants:
- Square-root staffing rule (Halfin-Whitt regime): for some quality-of-service constant
- Erlang A: add abandonment — customers leave after waiting too long; see Erlang A
- Skill-based routing: agents are heterogeneous; needs simulation
351.6. 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.
351.7. See also
- Erlang B — without queue (blocking)
- M/M/c — the queue model itself
- Erlang A — with abandonment
- Square-Root Staffing