# Little’s Law

A theorem that determines the average number of items in queuing systems

## What is Little’s Law?

Little’s Law is a theorem that determines the average number of items in a stationary queuing system based on the average waiting time of an item within a system and the average number of items arriving at the system per unit of time.

The law provides a simple and intuitive approach for the assessment of the efficiency of queuing systems. The concept is hugely significant for business operations because it states that the number of items in the queuing systems primarily depends on two key variables, and it is not affected by other factors such as the distribution of the service or service order.

Little’s Law can only be used in queuing systems. Almost any queuing system and even any sub-system (think about the single teller in a supermarket) can be assessed using the law. In addition, the theorem can be applied in different fields, from running a small coffee shop to the maintenance of the operations of a military airbase.

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### Origin of Little’s Law

Massachusetts Institute of Technology (MIT) professor John Little developed Little’s Law in 1954. The initial publication of the law did not contain any proofs of the theorem. However, in 1961, Little published proof that there is no such situation when the described relationship does not hold. Little later received recognition for his works in operations research.

### Formula for Little’s Law

Mathematically, Little’s Law is expressed through the following equation:

Where:

L – the average number of items in a queuing system

λ – the average number of items arriving at the system per unit of time

W – the average waiting time an item spends in a queuing system

### Example of Little’s Law

John owns a small coffee shop. He wants to know the average number of customers queuing in his coffee shop to decide whether he needs to add more space to accommodate more clients. Currently, his queuing area can accommodate no more than eight customers.

John measured that on average, 40 customers arrive at his coffee shop every hour. He also determined that on average, a customer spends around 6 minutes in a store (or 0.1 hours). Given the inputs, John can find the average number of the customers queuing in his coffee shop by applying the Little’s Law:

##### L  =  40 x 0.1  =  4 customers

The Little’s Law shows that on average, there are only four customers queuing in John’s coffee shop. Therefore, he does not require to create more space in the store to accommodate more queuing customers.