mathematics//probability//queueing theory//Little's law

Little's law is the theorem of queueing theory stating that in any stable system the mean number of items inside equals the mean arrival rate times the mean time each item spends inside, and it is used to get whichever of the three is hard to measure from the two that are easy.

Little's law is the theorem of queueing theory stating that in any stable system the mean number of items inside equals the mean arrival rate times the mean time each item spends inside, and it is used to get whichever of the three is hard to measure from the two that are easy.

L=λ WL=\lambda\,WL=λW

LLL is the mean number in the system, λ\lambdaλ the mean arrival rate and WWW the mean time from arrival to departure. If warehouse robots reach the charging area at 12 an hour and each stays 30 minutes, queueing and charging included, there are on average 6 robots in the area, and the floor plan needs room for 6 plus whatever the peaks demand.

Its strength is what it leaves out. It holds whatever the distribution of arrivals and service, whatever the order of service and however many servers there are, because it is an accounting identity: draw the number of items inside against time, and the area under that curve counts the same item-hours whether it is summed instant by instant (giving LLL) or item by item (giving WWW).

At a fixed throughput, time inside and stock inside are the same number seen twice.

A plant with 400 parts of work in process and an output of 100 parts a day has a lead time of four days, and the only way to shorten it without producing faster is to hold less in process.

In networks it is the bandwidth-delay product. A link carrying 100 Mbit/s with a round trip of 40 ms holds 4 Mbit (500 kB) in flight, which is the amount of window and buffer a sender needs to keep the link full.

It applies to any boundary drawn consistently: the waiting line alone (giving the mean queue length from the mean wait), the server alone, or a whole factory. Measurements of LLL, λ\lambdaλ and WWW must use the same boundary.

It speaks only of long-run averages in a stable system, where what enters eventually leaves. During a start-up, or an overload in which the line keeps growing, it does not hold, and it says nothing about variance or tails: two systems with the same LLL and WWW can have very different worst cases, which the M/M/1 queue and its relatives describe.