Answer first
What this calculator tells you
Calculate the average number of items in a system from the arrival rate and the time each spends there. Size a queue, a work-in-progress limit or a waiting room from two measured numbers. Formula: L = λ × W, the average number in the system equals the arrival rate times the average time each spends in it. At the worked-example inputs, the average number in the system is 15. Holding every other input steady, moving arrival rate (per hour) from 24 to 36 moves the result from 12 to 18.
Transparent method
The formula
Size a queue, a work-in-progress limit or a waiting room from two measured numbers.
Worked example
Example inputs
How to interpret the result
Little's law links three things in any stable system: the average number inside, the rate of arrivals and the average time each spends there. Thirty customers an hour who stay half an hour each keep about 15 people in the room at any moment. It holds for queues, clinics, factories and software backlogs, and it lets you find a third number from two you can measure.
At the worked-example inputs the average number in the system is 15. It rises with arrival rate (per hour) and average time in the system (hours).
These are planning metrics, not audited accounting or a valuation opinion.
Before you rely on it
What to check
Keep the units matched. An arrival rate per hour needs a time in hours, and mixing minutes with hours breaks the result.
The common error
Where people go wrong with little's law calculator
Using it on a system that is filling up or draining. The law describes a steady state, so a queue that grows all day is outside it.
Sensitivity evidence
How arrival rate (per hour) changes the average number in the system
Holding every other input at the worked-example value, moving arrival rate (per hour) from 24 to 36 moves the average number in the system from 12 to 18: a spread of 6, or 40% of the worked-example result.
| Arrival rate (per hour) | Average number in the system | Average time in the system (minutes) |
|---|---|---|
| 24 | 12 | 30 |
| 27 | 13.5 | 30 |
| 30worked example | 15 | 30 |
| 33 | 16.5 | 30 |
| 36 | 18 | 30 |
Every input, tested
Which input moves the average number in the system most
Of the 2 inputs, arrival rate (per hour) moves the average number in the system most (3 across the range tested) and average time in the system (hours) moves it least (3).
| Input | Tested from | To | Average number in the system at each end | Swing |
|---|---|---|---|---|
| Arrival rate (per hour) | 27 | 33 | 13.5 to 16.5 | 3 (20%) |
| Average time in the system (hours) | 0.45 | 0.55 | 13.5 to 16.5 | 3 (20%) |
Two variables at once
Average number in the system by arrival rate (per hour) and average time in the system (hours)
Across the grid the average number in the system runs from 9.6 to 21.6. Moving arrival rate (per hour) from 24 to 36 shifts it by 6 at the middle column, and moving average time in the system (hours) from 0.4 to 0.6 shifts it by 6 at the middle row, so neither is the bigger lever here.
| 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 |
The highlighted cell is the worked example: 15.
Step by step
The worked example, input by input
| Input | Value used | What it means |
|---|---|---|
| Arrival rate (per hour) | 30 | Enter the arrival rate (per hour) used in this calculation. |
| Average time in the system (hours) | 0.5 | Enter the average time in the system (hours) used in this calculation. |
| Average number in the system | 15 | |
| Average time in the system (minutes) | 30 | |
Inputs, definitions and assumptions
Arrival rate (per hour)
Enter the arrival rate (per hour) used in this calculation. The prefilled worked-example value is 30.
Average time in the system (hours)
Enter the average time in the system (hours) used in this calculation. The prefilled worked-example value is 0.5.
How to use this calculator
- 1Verify the inputs. Gather arrival rate (per hour) and average time in the system (hours) from your own documents; the prefilled values are examples.
- 2Save a baseline. The worked example puts the average number in the system at 15. Store your own version of it as Scenario A.
- 3Test one change. Start with arrival rate (per hour), the input with the biggest effect here: moving arrival rate (per hour) from 27 to 33 takes the average number in the system from 13.5 to 16.5, a swing of 20% of the worked-example figure.
- 4Check the extremes. At half the example arrival rate (per hour) (15) the average number in the system is 7.5; at double (60) it is 30.
People also ask
Frequently asked questions
How do you calculate little's law?
L = λ × W, the average number in the system equals the arrival rate times the average time each spends in it. At the worked-example inputs the average number in the system is 15.
What does the little's law result mean?
Size a queue, a work-in-progress limit or a waiting room from two measured numbers. At the worked-example inputs the average number in the system is 15. It rises with arrival rate (per hour) and average time in the system (hours).
How much does arrival rate (per hour) change the average number in the system?
Holding every other input at the worked-example value, moving arrival rate (per hour) from 24 to 36 moves the average number in the system from 12 to 18, a spread of 6.
What are the limits of this little's law calculator?
These are planning metrics, not audited accounting or a valuation opinion. The tables on this page test arrival rate (per hour) only from 24 to 36; a value outside that range is not tabulated here.
Which input moves the average number in the system most in the little's law calculator?
Ranked by how far each moves the average number in the system across the range tested: arrival rate (per hour) (3, 20%) and average time in the system (hours) (3, 20%).
If I double arrival rate (per hour) in the little's law calculator, does the average number in the system double?
Doubling it from 30 to 60 takes the average number in the system from 15 to 30, which is 2.00 times the worked-example figure. So the result scales almost exactly in proportion. Halving it to 15 gives 7.5.
How much does average time in the system (hours) matter in the little's law calculator?
The worked example uses 0.5. Holding every other input at its worked-example value, moving average time in the system (hours) from 0.45 to 0.55 takes the average number in the system from 13.5 to 16.5, a swing of 20% of the worked-example figure.
Which inputs change the average time in the system (minutes) in the little's law calculator?
At the worked-example inputs it is 30. Average time in the system (hours) takes it from 27 to 33.
How do I forecast next month's cash?
Start with cash on hand, add expected collections, subtract known payments and payroll, and adjust for seasonality. Update the forecast weekly, since small timing slips add up.
Sources and evidence
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