= Little's law
{c}
{title2=$L=\lambda_{\mathrm{eff}}W$}
{wiki}
= Little law
{c}
{synonym}
For a stable <queue> with finite long-run averages, a positive finite admitted rate, and negligible initial and terminal contributions after division by observation time, <Little's law> states
$$
L=\lambda_{\mathrm{eff}}W.
$$
Here $L$ is the time-average <queue population>, $\lambda_{\mathrm{eff}}$ is the admitted rate through the chosen system, and $W$ is the customer-average <customer sojourn time> inside that same boundary. Without abandonment, admitted rate equals completed-service <queue throughput> in this stable regime. It may differ from the offered arrival rate because of blocking. When customers abandon, count their admissions and exits too, and average their actual time inside; using only service completions would mismatch the population being measured.
For the waiting area alone the matching identity is
$$
L_q=\lambda_{\mathrm{eff}}W_q,
$$
where $W_q$ is mean <queueing delay>. For one server with no abandonment, service in arrival order, and finite mean <service time> $\mathbb E S$,
$$
W=W_q+\mathbb E S,\qquad L=L_q+\lambda_{\mathrm{eff}}\mathbb E S=L_q+\rho,
$$
with $\rho$ the <server utilization>. Mixing total population $L$ with waiting-only duration $W_q$ would count different portions of each customer's visit. Customers who start service immediately contribute zero to $W_q$, and must still be included in that customer average.
The strength of <Little's law> is that it does not require a <Poisson process>, an <exponential distribution> of <service times>, or service in arrival order. The <sample-path area identity for Little's law> explains why: counting customer-time across time slices or across customer visits gives the same area. A sufficient probabilistic setting is repeated <independent and identically distributed> empty-to-empty cycles with finite positive mean cycle duration, finite positive mean customer count, and finite mean population area; the <renewal-reward theorem> then supplies the long-run averages. Stability alone does not guarantee finite mean <queueing delay>: the <Pollaczek-Khinchine mean formula> shows why a finite <second moment> of <service time> matters for that mean in an <M G 1 queue>.
For example, suppose a service desk receives six requests per hour but admits only four; measured mean <customer sojourn time> of admitted requests is half an hour. Its mean <queue population> is $4\cdot(1/2)=2$ customers. Multiplying the offered rate by that duration would incorrectly give three. Conversely, counting two customers on average and measuring admitted <queue throughput> of four per hour gives $W=2/4=1/2$ hour. This lets one estimate a hard-to-track duration from occupancy and completed-job counts.
John D. C. Little published a rigorous proof in 1961 under finite-mean, <strict stationarity>, and <ergodicity> assumptions on the arrival process. Those were assumptions of that proof, rather than a requirement that every useful form of the identity have random arrivals. https://pubsonline.informs.org/doi/10.1287/opre.9.3.383[Little, A Proof for the Queuing Formula: L = λW, Operations Research 9(3), 383-387]{ref}. https://fisherp.scripts.mit.edu/wordpress/wp-content/uploads/2015/11/ContentServer.pdf[Original paper, opening statement]{ref}.
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