Little's law

In mathematical queueing theory, Little's result, theorem, lemma, law, or formula is a theorem by John Little which states that the long-term average number L of customers in a stationary system is equal to the long-term average effective arrival rate λ multiplied by the average time W that a customer spends in the system. Expressed algebraically the law is The relationship is not influenced by the arrival process distribution, the service distribution, the service order, or practically anything else. In most queuing systems, service time is the bottleneck that creates the queue. The result applies to any system, and particularly, it applies to systems within systems. For example in a bank branch, 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 be stable and non-preemptive; this rules out transition states such as initial startup or shutdown. In some cases it is possible not only to mathematically relate the average number in the system to the average wait but even to 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 the mean number in the system and the throughput are known, the average response time can be found using Little’s Law: mean response time = mean number in system / mean throughput For example: A queue depth meter shows an average of nine jobs waiting to be serviced.