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Little's Law Calculator

Enter the arrival rate and average time in system to find L.

How it is calculated

The tool multiplies two averages: L = λ × W. Here λ (lambda) is the average arrival rate, in items per unit of time, and W is the average time one item spends in the system, in the same time unit. L is the average number of items in the system at any moment.

The calculator does no unit conversion: if λ is per hour, W must be in hours, otherwise L is meaningless. Both inputs must be greater than zero; zero or negative values give no result. The law describes a stable system, where items leave as fast as they arrive on average. Results are shown to two decimals and need not be whole numbers, because L is an average. A line below the result repeats the multiplication with your numbers.

Worked example

A support desk receives 12 tickets per hour (λ = 12) and each ticket stays open for 2.5 hours on average (W = 2.5). Both figures use hours, so they can be multiplied directly: 12 × 2.5 = 30. The result is 30.00 items, so at any given moment there are about 30 open tickets. If the team wants fewer open tickets, it must cut the time each ticket spends in the system or reduce the arrivals.

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Common mistakes

  • Mixing time units, for example λ in items per minute and W in hours. Convert one of them first so both use the same unit.
  • Using the service time instead of the total time in the system. W includes waiting in the queue plus being served.
  • Entering the arrival count of a short busy period as λ. The law needs long-run averages of a stable system, not a peak.
  • Expecting a whole number. L is an average, so a result such as 30.00 or 7.50 is normal.
  • Rearranging the law incorrectly. If you know L and λ, then W = L ÷ λ; this tool only computes L.

About this calculator

Little's Law is a fundamental result in queueing theory stating that the average number of items in a stable system, L, equals the average arrival rate, λ, multiplied by the average time each item spends in the system, W — the elegant relationship L = λW. It holds regardless of the arrival pattern or how service time is distributed, which is what makes it so widely applicable.

This calculator takes the arrival rate and average time in system to compute L. It's used across operations management, software/systems engineering, and call center staffing — for example, to estimate how many items are sitting in a manufacturing queue, how many requests are in flight in a software system, or how many customers are typically in a store at once, just from throughput and dwell time.

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Comments (1)

  • Editorial team

    The law applies to any boundary you draw, so you can use it on just the waiting line. Enter the average wait before service as the time and the result counts only items still queued, not those being served. With 12 arrivals per hour and a 2.5-hour stay, the system holds 30.00 items on average. Pick the boundary before measuring, and use the same one for both inputs. What system are you modelling: tickets, orders, requests or something else?

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