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Little's law for stable flow systems

Many systems contain items that enter, spend some time inside and eventually leave. For a stable system observed over a sufficiently long interval, Little's law relates three long-run averages:

$$L=\lambda W,$$

where

  • $L$ is the average number of items in the system;
  • $\lambda$ is the average throughput or arrival/departure rate;
  • $W$ is the average time an item spends in the system.

Suppose a system completes an average of 30 items per hour and contains an average of 45 items between entry and completion. Then

$$W=\frac{L}{\lambda}=\frac{45}{30}=1.5\ \mathrm h.$$

The equation does not require every item to spend exactly 1.5 hours inside. It relates averages over time.

Little's law is remarkably general because it does not require a particular probability distribution for processing time. What matters is a consistent definition of the system boundary and sufficiently stable long-run averages.

In manufacturing, $L$ can represent work in process (WIP) and $W$ the average production flow time. In computing, the items might instead be requests inside a service. The same conservation-style relationship applies.

The law does not say that increasing the number of items inside a system increases throughput. If a limiting resource is already saturated, additional items can mainly increase waiting and therefore increase $W$.

Little's law is therefore a bookkeeping relationship connecting inventory, rate and time—not a claim that any one of them can be changed independently of system behavior.