Last updated: 2026-06-29

Little's Law

Little's law is one of the most widely used results in queueing theory. It ties together three quantities every performance engineer measures — how many items are in a system, how fast they arrive, and how long they stay — into a single relationship that holds for almost any stable system. The calculator solves the law in any direction: given two of the three quantities, it returns the third.

The relationship

For a system in steady state, the long-run average number of items present, $L$, equals the long-run average arrival rate, $\lambda$, multiplied by the average time an item spends in the system, $W$:

$L = \lambda \, W$

The remarkable part is how few assumptions it needs. John Little's 1961 proof requires only that the system be stable — that over a long enough window, items leave as fast as they arrive — and that the averages exist. It does not care whether arrivals are random or bursty, whether service times are constant or highly variable, or whether the queue is served first-in-first-out or by priority.

Reading it three ways

The same equation answers three different questions, which is why the calculator offers a mode for each:

• Concurrency from throughput and latency: $L = \lambda W$. A service handling 2,000 requests per second, each taking 50 ms, holds $2000 \times 0.05 = 100$ requests in flight on average.
• Throughput from concurrency and latency: $\lambda = L / W$.
• Latency from concurrency and throughput: $W = L / \lambda$.

This makes the law a quick consistency check on measurements and a tool for sizing pools — connection pools, thread pools, or worker counts — to match a target load.

Keeping units consistent

The only rule is that the rate and the time share a time unit. If $\lambda$ is in requests per second, then $W$ must be in seconds and $L$ comes out as a pure count. If you prefer to think in requests per hour, measure $W$ in hours.

Limits

Little's law describes long-run averages, so it is silent about momentary spikes, tail latency, and variance. It also assumes stability: if the backlog keeps growing because arrivals outpace service, the average number in the system is not well defined and the law no longer applies. For a model that captures waiting time and queue length under random arrivals, see the M/M/1 Queue Calculator.