Erlang Law of Averages

The Erlang Law of Averages holds when volume stays in a near-linear range of the staffing curve (the gap between average-of-staffing and staffing-of-average is small) and breaks across the convex low-volume region or with extreme peaks.

Erlang Law of Averages is the practical result that, over a planning horizon, the average staffing requirement can be approximated by computing the staffing for the average volume — rather than calculating required agents interval by interval and averaging the results — with little loss of accuracy under the right conditions. It is a useful shortcut for long-range capacity planning in workforce management, and it sits in deliberate tension with the flaw of averages, which warns that the function of an average is generally not the average of the function. The Erlang Law of Averages is best understood as a statement of when that gap is small enough to ignore.

The apparent contradiction

The flaw of averages says that for a nonlinear relationship, planning on a single average input misstates the result: required staffing for average volume need not equal the average required staffing across variable volume. Taken literally, that seems to forbid deriving period staffing from average volume. The resolution is that the size of the error depends on how curved the staffing relationship is over the relevant volume range — by Jensen's inequality, the gap grows with curvature and with the spread of volume. Where the relationship is nearly straight over the operating range, the gap is negligible and the shortcut is safe; where it bends sharply, the shortcut fails.

Why it usually holds for staffing

Required staffing is, for most of its range, only mildly nonlinear in volume. The square-root staffing law expresses required agents as the offered load plus a safety cushion proportional to the square root of the load: N ≈ R + β√R. The first term is linear in volume, and only the √ term curves — gently, and less so as volume grows. So over a moderate-to-high volume range the staffing curve is close to linear, the average-of-staffing and the staffing-of-average nearly coincide, and the Erlang Law of Averages holds to good approximation. This is why long-term planners can size quarterly or annual FTE from average volume without summing thousands of interval-level calculations. The behaviour was cited as a supported capability in CCmath's 2026 "Erlang v4" engine release.

When it breaks

The approximation degrades exactly where the staffing curve is most nonlinear or volume most variable:

• Very low volumes. At small offered loads the √ term dominates and the curve bends sharply — the regime of the staffing cliff — so averaging across low-volume intervals misstates the requirement. Accurate low-volume results depend on careful interpolation of the Erlang curve.
• Extreme peaks and high volatility. When volume spans a wide range within the period, the chord across the curve sits well above the curve at the mean, and staffing from average volume understates the requirement — the flaw of averages reasserting itself.
• When the question is peak service, not average staffing. The law concerns the average staffing level over a horizon. It says nothing about whether individual peak intervals are covered; interval-level execution still requires interval-level staffing and percentile protection.

Practical use

• Long-range planning. Use average volume to size budgets, hiring plans, and FTE over quarters or years, where the average staffing level is what matters and the law holds well.
• Keep interval math for execution. Use interval-level Erlang for the schedule period, where peaks and the staffing cliff matter.
• Check the regime. Before relying on the shortcut, confirm volume stays in a moderate, near-linear range; for low-volume or highly peaked operations, fall back to interval computation.

Maturity Model Position

In the WFM Labs Maturity Model™, knowing when the averaging shortcut is safe is a mark of staffing-math judgment.

• Level 1–2 (Emerging / Foundational) — average volume is used for planning without awareness of when it misleads, or interval math is run everywhere without recognizing where averaging would have been adequate.
• Level 3 (Progressive) — planners use average-volume staffing for long-range aggregates and interval-level staffing for execution, and know the law breaks at low volume and high peaks.
• Level 4–5 (Advanced / Pioneering) — the choice between averaged and interval-level computation is made deliberately by regime, and tooling applies the appropriate method automatically.

See also

• The Flaw of Averages
• Square Root Staffing Law
• Fractional Agents and Staffing Interpolation
• Capacity Planning Methods
• Staffing to Percentile vs Mean Forecast
• Erlang C
• CCmath