353. Square-Root Staffing
A simple rule of thumb for sizing call-center / service-system capacity. Halfin-Whitt asymptotic regime (1981).
353.1. The rule
Given offered load Erlangs, optimal number of servers is approximately:
— a base of servers (deterministic capacity) plus a safety margin that grows as .
353.2. Three quality regimes (parameterized by )
- Efficiency-driven (): servers saturated, customers wait often. Wait time scales as .
- Quality-driven ( or so): servers idle often, customers rarely wait. Wait time scales as with very small constant.
- Quality-and-Efficiency-Driven (QED, ): balanced — wait probability and idle probability both moderate.
For typical call centers targeting “80% of calls answered within 20 sec”, .
353.3. Why ?
In the heavy-traffic limit, queueing behavior is governed by central limit-style fluctuations. The natural fluctuation scale of an offered load is — same as the standard deviation of a Poisson() distribution.
If you provision exactly servers, fluctuations of size overwhelm capacity routinely. If you provision , you absorb typical fluctuations. Bigger → more safety.
353.4. Example
Call center: 50 calls/min, average 3 min per call → Erlangs.
| Strategy | Behavior | ||
|---|---|---|---|
| No safety | 0 | 150 | 50% chance customer waits |
| Modest | 0.5 | 156 | 30% wait probability |
| Quality | 1.5 | 168 | 10% wait probability |
| Premium | 2.5 | 181 | < 3% wait probability |
(Wait probabilities approximate, depend on abandonment, service distribution, etc.)
353.5. Practical use
- Estimate offered load
- Choose target service level (e.g., 80% answered in 20 sec)
- Look up corresponding (or simulate to calibrate)
- Staff
Compare to Erlang C / Erlang A exact calculations — typically very close.
353.6. Why it’s the right scale
For a fixed fraction-of-time-waiting target (constant probability of waiting), stays constant as offered load grows. So if you double traffic ( doubles), safety margin only grows as x — capacity grows by less than linearly.
This is the economies of scale of pooling: big call centers need proportionally less slack capacity than small ones.
353.7. See also
- Erlang C — exact wait probability
- Erlang B — no queue / blocking
- Erlang A — with abandonment
- Sakasegawa — G/G/c approximation