Little's Law
A mathematical theorem stating that the average number of items in a system equals the average arrival rate multiplied by the average time each item spends in the system.
Also known as: Little's Theorem, Little's Formula, L = λW
Category: Principles
Tags: lean, kanban, metrics, operations, mathematics, productivity
Explanation
Little's Law is a fundamental theorem in queueing theory, proven by John Little in 1961. It states:
**L = λ × W**
Where:
- **L** = Average number of items in the system (WIP)
- **λ** (lambda) = Average arrival rate (throughput, when the system is stable)
- **W** = Average time an item spends in the system (lead time)
In workflow terms: **WIP = Throughput × Lead Time**
**Why It's Powerful**:
Little's Law is remarkably general. It applies to any stable system — manufacturing lines, software development queues, hospital emergency rooms, supermarket checkout lines, or email inboxes. The only requirement is that the system is in a steady state (arrivals roughly equal departures over time).
**Practical Implications**:
Rearranging the formula reveals actionable insights:
| Formula | Insight |
|---------|---------|
| **Lead Time = WIP ÷ Throughput** | To reduce lead time, either reduce WIP or increase throughput |
| **Throughput = WIP ÷ Lead Time** | Throughput depends on how much is in progress and how fast it moves |
| **WIP = Throughput × Lead Time** | More WIP means longer lead times (at constant throughput) |
**The WIP Trap**:
Many teams try to increase throughput by starting more work (increasing WIP). Little's Law shows why this backfires: if throughput stays the same (because the bottleneck hasn't changed), adding WIP directly increases lead time. Each item takes longer, and customers wait more.
**Applications in Knowledge Work**:
1. **Kanban WIP limits**: Justified mathematically by Little's Law — limiting WIP reduces lead time
2. **Sprint planning**: If a Scrum team's throughput is 10 items/sprint, committing to 15 items means lead time exceeds one sprint
3. **Personal productivity**: If you have 30 tasks (WIP) and complete 5 per week (throughput), your average lead time is 6 weeks
4. **Hiring decisions**: Adding people doesn't immediately increase throughput, so it doesn't immediately reduce lead time
**Assumptions for Validity**:
- The system is in **steady state** (average arrival rate ≈ average departure rate)
- All items that enter the system eventually leave
- WIP is measured consistently
- Averages are calculated over the same time period
When these assumptions hold, Little's Law is exact — not an approximation. It's one of the few mathematical certainties in process management.
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