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37

Performance

Little's Law

Queueing Invariant.

The article

If

In any stable system, the average number of items in the system (L) equals the average arrival rate (λ) times the average time each item spends in it (W): L = λW. It holds regardless of arrival distribution or service discipline.

Then

“To cut latency (W) without adding capacity, cut work-in-progress (L). Cap concurrent requests — a shorter queue is faster than a faster server.”

Unless

Backpressure / Admission Control

A queue with arrival rate lambda, L items in the system, and time W spent in itArrivals λL items in the systemDeparturesW — time each item spends insideL = λWcap L → W falls, with λ unchanged
L = λW binds the three quantities, so you cannot move one without moving another. Arrival rate is set by your users, not by you. That leaves one lever you actually control: cap the work in progress, and latency falls out of the arithmetic — no faster server required.

Source

John D. C. Little, A Proof for the Queuing Formula L = λW (Operations Research, 1961).

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See also

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