Walk into a coffee shop. Count the people inside: that’s \(L\). Stand at the door and count how many people walk in per minute: that’s \(\lambda\). On average, each person stays \(W\) minutes.
Little’s law says \(L = \lambda W\), and it holds for almost any system in steady state — no assumptions about distributions, queue discipline, or service order. It just works. Factories, web servers, hospital beds, highway traffic. If the system is stable, the law applies.
formula
\[L = \lambda W\]where:
- \(L\) — long-term average number of items in the system
- \(\lambda\) — long-term average arrival rate (items per unit time), which in steady state equals the throughput
- \(W\) — long-term average time an item spends in the system
Rearranged:
- \(W = \frac{L}{\lambda}\) — average time in system
- \(\lambda = \frac{L}{W}\) — throughput
notes
Remarkably general. The law is distribution-free. It doesn’t depend on arrival patterns, service time distributions, number of servers, or queue discipline (FIFO, LIFO, random). Any system in steady state.
Steady state is the real requirement. The system must be stationary: the average arrival rate must equal the average departure rate over the observation window. Everything that enters must eventually leave. If work-in-progress accumulates unboundedly, the law doesn’t apply.
Averages, not snapshots. \(L\), \(\lambda\), and \(W\) are all long-run time-averages. At any given instant, the relationship may not hold. This is where practitioners trip up — measuring over short windows where the system isn’t in steady state.
Stronger than you’d think. Little’s original 1961 proof assumed stationarity and ergodicity. Stidham (1972) proved a purely deterministic version, showing the law is even more general than the original probabilistic assumptions suggested.