Little’s Law Calculator
Queueing theory: the average number in a system equals arrival rate times average time in the system, L = λ · W.
Inputs
Drag the handle (or focus it and press ↑/↓) to reorder. Scientific notation such as 1.5e-3 is accepted. Pick any unit; conversion is automatic. In Advanced you can solve for L, λ or W. Pick any unit; conversion is automatic.
Results
Enter values and press Calculate. Your answer will appear here.
Items in system vs. time in system
For a fixed arrival rate, L rises in direct proportion to W. The solid line is your λ, the dashed lines are half and double that rate, and the dot is your result.
What is being calculated
Common examples
| Case | λ | W | L |
|---|
Illustrative inputs, not manufacturer data.
L for λ = 10 per hour at different W
| W | L = λ · W |
|---|
Doubling the time in the system doubles the number of items in it, if arrivals stay the same.
Unit conversions
| Quantity | Conversion |
|---|---|
| Rate | 1 /min = 60 /h = 1/60 /s · 1 /h = 24 /day · 1 /s = 3,600 /h |
| Time | 1 h = 60 min = 3,600 s · 1 day = 24 h · 1 s = 1,000 ms |
| Consistency rule | use matching time units: 20 /h × 0.25 h = 5 |
Formula
- L: average number of items in the system
- λ: average arrival rate (items per unit time)
- W: average time an item spends in the system
Example: a coffee shop sees λ = 20 customers per hour and each stays W = 15 min = 0.25 h. Then L = 20 × 0.25 = 5 customers on average. Rearranged: λ = L / W and W = L / λ.
Little’s Law holds for long-run averages in a stable system (what comes in eventually goes out), regardless of the arrival pattern, the service pattern or the queue discipline. It also applies to any part of the system: for the queue alone Lq = λ·Wq, and for the servers Ls = λ·Ws. Advanced mode uses this to split L and W, and reports utilisation ρ = λ·Ws / c. If ρ reaches 100 % the system is unstable and the law no longer gives a steady-state answer. Use the same unit of time for λ and W, which this calculator handles for you.
References
- Wikipedia, Little’s law: statement, assumptions and examples
- J. D. C. Little (1961), A Proof for the Queuing Formula: L = λW: the original proof, Operations Research
- Wikipedia, Queueing theory: arrival rate, service time and utilisation
How to Use This Tool
- Pick a preset, or enter the arrival rate λ and its unit.
- Enter the average time in the system W and its unit.
- In Advanced, choose to solve for λ or W instead, or add a service time and number of servers.
- Click Calculate.
- Read the result and the other two quantities, then check the chart.
What This Tool Offers
- Basic and Advanced modes on one calculation engine
- Rate and time unit conversion, from milliseconds to days
- Solve for L, λ or W (Advanced)
- Queue and service split, utilisation and a stability warning (Advanced)
- L-versus-W chart and labelled system-flow diagram
- Reorderable inputs, presets and a mobile-friendly layout
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| L | Average number in system | |
| λ | Arrival rate | items/sec |
| W | Average time in system | sec |
What it means
Little’s law is a fundamental theorem in queuing theory, applicable to any stable system where arrivals and departures are balanced. It states that the long‑term average number of customers (or jobs) in a system L equals the average arrival rate λ (customers per unit time) multiplied by the average time W that a customer spends in the system. The relationship holds regardless of the service time distribution, queue discipline, or number of servers. This makes it a powerful tool for performance analysis of computer systems, networks, call centers, and manufacturing. For example, in a web server, if requests arrive at 100 req/s and the average response time is 0.05 s, then the average number of concurrent requests in the system is L = 100 × 0.05 = 5. Little’s law can be applied to any subsystem (e.g., the queue itself or the server) and is often used to compute required system capacity or to infer one variable when the other two are known. It is also used in agile project management (e.g., lead time vs. work in progress) and operations research.
Worked example
Little's Law – Two Examples
Real‑World| Parameter | Value |
|---|---|
| λ (arrival rate) | 5 customers/min |
| W (waiting time) | 4 min |
| Parameter | Value |
|---|---|
| λ (requests/sec) | 300 |
| W (response time) | 0.05 s |
Common mistakes
- Arrival rate λ: Must be in the same time unit as W (e.g., customers per hour and hours).
- Steady state: Little’s law assumes the system is in steady state – it does not apply to transient conditions.
- Definitions: L is the average number of items in the system (including those in service and waiting).
- Conservation: Little’s law holds for any queuing system, regardless of the distribution of arrivals or service times.
- Multiple classes: For multiple classes of customers, the law applies to each class separately or to the aggregate.
Applications
Little's law is a simple yet powerful theorem in queuing theory, relating the average number of items in a stable system (L) to the average arrival rate (λ) and the average time an item spends in the system (W). It applies to any queuing system, from computer networks and call centres to manufacturing and service operations. In computing, it is used to analyse performance metrics such as the average number of requests in a web server, the latency of packet queues, and the capacity of multi‑threaded systems. Little's law enables engineers to derive one variable if the other two are known, facilitating capacity planning, bottleneck identification, and service level agreement (SLA) design. Its simplicity and generality make it a cornerstone of performance engineering.
- Performance analysis of web servers and databases
- Design of call centres and customer service queues
- Network traffic engineering and packet queuing
- Manufacturing and production line optimisation
- Capacity planning and resource allocation
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