Little’s law

Little’s law is a theorem from queueing theory: the average number of items in a system, L, equals the average arrival rate, λ, multiplied by the average time an item spends in the system, W – that is, L = λW. It was proven rigorously by the mathematician John Little in 1961, though the relationship had been used informally before then.

The theorem holds for any stable system regardless of the details of arrival patterns or service times, which is what makes it useful as a general planning tool rather than a model specific to one kind of process. In Kanban and lean software development, Little’s law is the mathematical basis for the claim that limiting work in progress shortens lead time. For a fixed rate at which work items complete, reducing the number of items in flight at any one time proportionally reduces how long each item takes to get through the system. This is the reasoning behind Kanban’s insistence on explicit work-in-progress limits.

Little’s law also underlies capacity planning in distributed systems: the relationship between latency, throughput, and the number of concurrent requests a system must be able to hold follows the same L = λW form, and is used to reason about how many concurrent workers or connections a system needs to sustain a target throughput at a target latency.

See also

• Laws and principles

References

• Little, John D. C. (1961). A Proof for the Queuing Formula: L = λW. Operations Research, 9(3), 383–387.