Little's Law: worked example
| Arrival rate (per hour) | 30 |
|---|---|
| Average time in the system (hours) | 0.5 |
| Average number in the system | 15 |
| Average time in the system (minutes) | 30 |
Little's Law: formula
John D. C. Little (1961): A Proof for the Queuing Formula · MIT: Queueing theory and Little’s formula
Common error: Using service time alone while the item count includes both queued and in-service work. Match the time measurement to the same entry and exit boundary used for the count.
Measure the average arrivals per hour and average hours spent inside one defined process boundary. Use a representative stable interval for both measurements. If the count includes waiting and processing, the time measurement must cover that same entry-to-exit span.
Little's Law: changing Arrival rate (per hour)
| Arrival rate (per hour) | Average number in the system | Average time in the system (minutes) |
|---|---|---|
| 24 | 12 | 30 |
| 27 | 13.5 | 30 |
| 30 (example) | 15 | 30 |
| 33 | 16.5 | 30 |
| 36 | 18 | 30 |
Little's Law: compare input ranges
| Input | Tested from | To | Average number in the system: low input | Average number in the system: high input | Swing |
|---|---|---|---|---|---|
| Arrival rate (per hour) | 27 | 33 | 13.5 | 16.5 | 3 (20%) |
| Average time in the system (hours) | 0.45 | 0.55 | 13.5 | 16.5 | 3 (20%) |
Little's Law: individual input comparisons
| Input | Value | Average number in the system | Average time in the system (minutes) |
|---|---|---|---|
| Arrival rate (per hour) | 24 | 12 | 30 |
| 27 | 13.5 | 30 | |
| 30 | 15 | 30 | |
| 33 | 16.5 | 30 | |
| 36 | 18 | 30 | |
| Average time in the system (hours) | 0.4 | 12 | 24 |
| 0.45 | 13.5 | 27 | |
| 0.5 | 15 | 30 | |
| 0.55 | 16.5 | 33 | |
| 0.6 | 18 | 36 |
Little's Law: combined input changes
| Arrival rate (per hour) / Average time in the system (hours) | 0.4 | 0.5 | 0.6 |
|---|---|---|---|
| 24 | 9.6 | 12 | 14.4 |
| 27 | 10.8 | 13.5 | 16.2 |
| 30 | 12 | 15 | 18 |
| 33 | 13.2 | 16.5 | 19.8 |
| 36 | 14.4 | 18 | 21.6 |
Little's Law: which assumptions move the result?
Little's Law: values and definitions
| Quantity and definition | Value used |
|---|---|
| Arrival rate (per hour) | 30 |
| Average time in the system (hours) | 0.5 |
| Average number in the system | 15 |
| Average time in the system (minutes) | 30 |
Little's Law: half and double Arrival rate (per hour)
| Arrival rate (per hour) | Average number in the system |
|---|---|
| 15 | 7.5 |
| 30 | 15 |
| 60 | 30 |
The original theorem uses finite averages and stationarity assumptions. A short observation during buildup or drainage need not represent those long-run quantities. Arrival-pattern variability and service-time variability can produce peaks that the average identity does not determine.
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Little’s identity relates average items present, average arrival rate and average time inside the same measured boundary. The assumed 30 arrivals per hour and half-hour mean duration give 15 items on average; a single observation need not equal that average.
Keep the units matched. An arrival rate per hour needs a time in hours, and mixing minutes with hours breaks the result.
Use the takt time calculator to set the pace, and the overall equipment effectiveness page for equipment use. The safety stock and reorder point calculators apply the same idea to inventory. In a service setting, compare with the staffing you have.
Can I use it for inventory?
Yes. Average inventory equals the rate of demand times the average time items sit in stock, which links to inventory days.
What is Little's law?
A relationship stating that the average number of items in a system equals the average arrival rate multiplied by the average time each item spends there.
Does it require constant arrivals?
No. It holds for any stable system, whatever the arrival pattern, as long as you use long-run averages.