Kingman's Formula
A queueing theory approximation showing that waiting time grows explosively as utilization approaches 100%, amplified by variability in arrivals and service times.
Also known as: Kingman Formula, Kingman's Equation, VUT Equation, Kingman Approximation
Category: Principles
Tags: mathematics, operations, lean, productivity, flow
Explanation
Kingman's Formula, published by John Kingman in 1961, approximates the average waiting time in a queue. In simplified form:
Wait time ≈ (u / (1 - u)) × ((ca² + cs²) / 2) × average service time
where u is utilization, and ca and cs are the coefficients of variation of arrival times and service times.
## What It Says
The formula has three parts, and two of them are bad news for busy teams:
- **Utilization term u / (1 - u)**: At 50% utilization it equals 1. At 80%, 4. At 90%, 9. At 95%, 19. Waiting time does not grow linearly with load: it explodes as you approach full capacity.
- **Variability term**: The more unpredictable the arrival of work and the time it takes, the longer the waits. Knowledge work has very high variability.
- **Service time**: The basic time to process an item.
## Why It Matters for Knowledge Work
- **Fully loaded people and teams are slow**: If everyone is 100% busy, new work waits a long time. This is the mathematical core of the resource efficiency trap.
- **Slack is not waste**: Spare capacity is what allows work to flow and urgent items to be handled quickly.
- **Reduce variability**: Smaller and more uniform work items (small batches) shrink queues as much as adding capacity does.
- **Combine with Little's Law**: Little's Law says lead time = WIP / throughput. Kingman explains why WIP and lead times balloon when utilization is high.
## Practical Implications
- Plan for 70-80% utilization, not 100%.
- Limit work in progress.
- Break work into smaller, more predictable pieces.
- Leave buffer time in your calendar for the unexpected rather than scheduling every hour.
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