A general relation in queueing systems stating that, in steady state for a stable system with conserved flow, the long‑run time‑average number of items L in the system equals the long‑run average arrival rate λ times the long‑run average time W an item spends in the system: L = λ W. It applies to averages taken over the same population and time frame and does not require specific arrival or service distributions.

Principle

Conservation of flow in steady state links population, throughput and delay: average inventory equals throughput times average residence time, independent of service discipline or interarrival/service distributional forms provided stability and well‑defined time averages hold.

Demonstration

Illustrative scenario → A call center receives calls at long‑run average rate λ calls/hour; each call spends on average W hours in the system (waiting plus service). Recognition → System has reached statistical steady state and averages are well defined. Action → Compute L via measurement or apply L=λW. Consequence → The average number of concurrent calls L equals λ·W; reducing W (faster service) or reducing λ (less arrivals) lowers average load.

Misapplication

Applying L=λW to transient periods, to systems with losses where arrivals are not conserved (e.g., blocked/lost customers) without using effective throughput, or confusing instantaneous counts with long‑run averages — the error is mismatching the averaging window or ignoring unstable/ill‑defined averages.

Consequence

Provides a robust accounting identity used in capacity planning, performance benchmarking and back‑of‑envelope estimates across many systems: adjusting throughput or delay predicts changes in average population, independent of distributional details.

Reversal

If the system is unstable (arrival rate exceeds service capacity), if averages are not well defined, or when items are removed before entering (lossy systems) without using effective arrival rate, Little's Law does not apply as stated; one must condition on the subsystem where conservation holds or use appropriate effective rates.

Boundary

Within: stable queueing systems in steady state with conserved flow and well‑defined long‑run averages; applicable to subsystems where these conditions hold. Boundary case: systems with balking, reneging or blocking require careful use of effective throughput and consistent averaging. Outside: transient analysis windows without stationarity, unstable systems, or ill‑posed averaging.

Semantic Tension

Tension between the universality of Little's conservation identity (insensitive to distributional form) and the need for detailed stochastic analysis to predict variability and tail behavior — Little's Law gives mean relations but not dispersion or performance guarantees about variability.

Synthesis

Little's Law is a powerful, distribution‑free conservation relation for steady‑state averages: it yields immediate mean relations useful for planning and sanity checks but must be paired with other queueing analyses to address variability, transients, losses or instability.