Probability theory and stochastic processes Special processes Theorem

Little's law

Statement

Consider a system, empty at time $0$ , that customers enter at times $0 \le a_1 \le a_2 \le \cdots$ and leave at times $d_k \ge a_k$ , with $a_k \to \infty$ . Let $W_k = d_k - a_k$ be the time customer $k$ spends in the system, let $A(t) = \#\set{k : a_k \le t}$ , and let $L(t) = \#\set{k : a_k \le t < d_k}$ be the number in the system at time $t$ . Assume that the limits $\lambda = \lim_{t \to \infty} \frac{A(t)}{t} \quad \text{and} \quad W = \lim_{n \to \infty} \frac1n \sum_{k=1}^n W_k$ exist and are finite. Then the limit $L = \lim_{t \to \infty} \frac1t \int_0^t L(s)\, ds$ exists and $L = \lambda W.$

Remarks

The proof compares two ways of computing the total time spent in the system up to time $t$ . The integral $\int_0^t L(s)\, ds$ lies between the sum of the $W_k$ over customers who have left by time $t$ and the sum over customers who have arrived by time $t$ . Dividing by $t$ , both bounds tend to $\lambda W$ ; the upper bound needs the fact that $W_n / a_n \to 0$ , which follows from the two hypotheses. Example: in the stable M/M/1 queue with $\lambda = 3$ and $\mu = 4$ , the mean number in the system is $L = 3$ , so the mean time in the system is $W = L / \lambda = 1$ . The law needs no assumption on the arrival law, the service law, the number of servers or the order of service.

Going deeper

For a student

In queueing theory one usually computes one of $L$ and $W$ from a model and gets the other one free. For the M/M/1 queue, the stationary law gives $L = \rho/(1 - \rho)$ , and Little's law gives $W = 1/(\mu - \lambda)$ . Applied to the waiting room alone, it links the mean queue length and the mean waiting time before service. In the stochastic setting it holds for stationary systems with expectations in place of time averages.

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M/M/1 queue example
Queueing theory concept
Little's law statement

Tag	`JN2YME6F4A`
URI	`m2://JN2YME6F4A`
Kind	statement, theorem
Writing	W0 stub
MSC	60K25, 90B22
Status	This copy of the site has no build data. Run `m2 status JN2YME6F4A`.
About	Queueing theory
Uses	Queueing theory, M/M/1 queue
Same as	Wikidata Q617388 Wikipedia Little's law
Changed	2 November 2026