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  "body": "\n## Statement\n\nConsider 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 $a_k \\to \\infty$.\nLet $W_k = d_k - a_k$ be the time customer $k$ spends in the system, let $A(t) = \\#\\set{k : a_k \\le t}$, and let $L(t) = \\#\\set{k : a_k \\le t \u003c d_k}$ be the number in the system at time $t$.\nAssume that the limits\n$$\\lambda = \\lim_{t \\to \\infty} \\frac{A(t)}{t} \\quad \\text{and} \\quad W = \\lim_{n \\to \\infty} \\frac1n \\sum_{k=1}^n W_k$$\nexist and are finite.\nThen the limit $L = \\lim_{t \\to \\infty} \\frac1t \\int_0^t L(s)\\, ds$ exists and\n$$L = \\lambda W.$$\n\n## Remarks\n\nThe proof compares two ways of computing the total time spent in the system up to time $t$.\nThe 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 $t$.\nDividing 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.\nExample: in the stable [[M/M/1 queue|GTFRPF5J2K]] with $\\lambda = 3$ and $\\mu = 4$, the mean number in the system is $L = 3$, so the mean time in the system is $W = L / \\lambda = 1$.\nThe law needs no assumption on the arrival law, the service law, the number of servers or the order of service.\n\n## Explanations\n\n### L1 human\n\nAverage number of customers inside equals arrival rate times average time each one stays.\nIf $10$ people per minute enter a museum and each stays $30$ minutes, there are about $300$ people inside.\nIt is a piece of accounting, so it holds for almost any system with a stable long-run behaviour.\n\n### L2 human\n\nIn [[queueing theory|P64V0HTE78]] one usually computes one of $L$ and $W$ from a model and gets the other one free.\nFor the M/M/1 queue, the stationary law gives $L = \\rho/(1 - \\rho)$, and Little's law gives $W = 1/(\\mu - \\lambda)$.\nApplied to the waiting room alone, it links the mean queue length and the mean waiting time before service.\nIn the stochastic setting it holds for stationary systems with expectations in place of time averages.\n",
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