Little's law

In queueing theory, a discipline within the mathematical theory of probability, Little's result, theorem, lemma, law or formula is a theorem by John Little which states:

Although it looks intuitively reasonable, it is quite a remarkable result, as the relationship is "not influenced by the arrival process distribution, the service distribution, the service order, or practically anything else."

The result applies to any system, and particularly, it applies to systems within systems. So in a bank, the customer line might be one subsystem, and each of the tellers another subsystem, and Little's result could be applied to each one, as well as the whole thing. The only requirements are that the system is stable and non-preemptive; this rules out transition states such as initial startup or shutdown.

In some cases it is possible to mathematically relate not only the average number in the system to the average wait but relate the entire probability distribution (and moments) of the number in the system to the wait.

In a 1954 paper Little's law was assumed true and used without proof. The form L = λW was first published by Philip M. Morse where he challenged readers to find a situation where the relationship did not hold. Little published in 1961 his proof of the law, showing that no such situation existed. Little's proof was followed by a simpler version by Jewell and another by Eilon. Shaler Stidham published a different and more intuitive proof in 1972.

Imagine an application that had no easy way to measure response time. If you can find the mean number in the system and the throughput, you can use Little's Law to find the average response time like so:

For example: A queue depth meter shows an average of nine jobs waiting to be serviced. Add one for the job being serviced, so there is an average of ten jobs in the system. Another meter shows a mean throughput of 50 per second. You can calculate the mean response time as: 0.2 seconds = 10 / 50 per second. When exploring Little’s law and learning to trust it, be aware of the common mistakes of using arrivals (work arriving) when throughput (work completed) is called for and not keeping the units of your measurements the same.

...
Wikipedia