# Little’s Law

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

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## 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 system primarily depends on two key variables and is not affected by other factors, such as the distribution of the service or service order.

Almost any queuing system and even any sub-system (think about a 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 proof of the theorem. However, in 1961, Little published proof that there is no queuing situation where the described relationship does not hold. Little later received recognition for his work 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 customers. Currently, his queuing area can accommodate no more than eight people.

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 his store (or 0.1 hours). Given these inputs, John can find the average number of customers queuing in his coffee shop by applying Little’s Law:

##### L  =  40 x 0.1  =  4 customers

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