# Little law

Little's law , also known as Little's theorem , Little's theorem , or Little's formula , is an important law in queuing theory . It was formulated and proven by John DC Little in 1961 .

Little's Law states that the average number of customers in a waiting system that is in a stable state is equal to the product of their average arrival rate and their average length of stay in the system . ${\ displaystyle L}$${\ displaystyle \ lambda}$${\ displaystyle W}$

${\ displaystyle L = \ lambda W}$

Although this makes intuitive sense, it is a remarkable result: It implies that this behavior is completely independent of the probability distributions used and therefore no assumptions about the distribution of arrival times or the handling discipline have to be made. The average waiting time with FIFO is just as great as with LIFO .

Little's law applies not only to an isolated operator station, but also to networks made up of waiting systems. For example, in a bank you can view the queue of a single counter as a subsystem and each additional counter as an additional subsystem. Little's law can be applied both to the subsystems individually and to the entire system. The only condition is that the system is stable - it must not be in a transition stage (start, end phase).

## example

A machine work station or service counter at which only one workpiece or customer order can be processed at the same time is to be set up. The work or service area includes a waiting area for newly arriving workpieces or customers. The average lead time is calculated as the sum of waiting and operating time . The arrival rate is known. Little's law can be used to determine the average number of workpieces or customers in the overall system or only in the waiting area . With these results, for example, the size of the waiting area can be dimensioned accordingly. The still missing parameter can be calculated in this constellation with the help of the formula for an M / M / 1 waiting system. ${\ displaystyle W = W_ {Q} + W_ {S}}$${\ displaystyle \ lambda}$${\ displaystyle L = \ lambda W}$${\ displaystyle L_ {Q} = \ lambda W_ {Q}}$

## literature

• Little, JDC: A Proof of the Queuing Formula L = λ W. In: Operations Research. 9, 1961, 383-387, JSTOR 167570 .
• Arnold, Dieter; Furmans, Kai: Material flow theory in logistics systems . 5th enlarged edition. Berlin, Heidelberg: Springer-Verlag, ISBN 978-3-540-45659-9 .