## What is **Little’s Law in Queuing Theory**

Do you want to improve productivity? Then “Little’s Law in Queuing Theory” is for you.

Queueing theory is the analysis of queues, or in other words, it is the study of lines. The aim of this study is to find ways to make long lines in front of a bank or an ATM move quicker.

Little’s Law in Queuing Theory is an equation showing that the average number of customers in a queueing system. That is equal to their average rate multiplied by the average amount of time they spend in the system. In other words:

“Number of items in the queue” = “arrival rate” x “average time spent in the queue”

This law applies only to queueing systems. These are systems in which work has to enter and leave and the work never ends while inside.

Recall standing in a queue, the point is never to queue but you enter the queue, spend time going through the process of queueing. And then finally move to the front to do whatever you were waiting for in the line. Another example is a production or assembly line. Sure, the work done is faster and more efficient but still, products wait in a queue to enter a production unit, and then leave for the next step. The work being done while in the queue doesn’t matter.

**HISTORY OF THE LAW**

The law itself is named after John Little – an MIT professor who first mathematically proved the law in 1961. The law existed beforehand, but until Little, there wasn’t a set mathematical definition of it or proof for its validity.

Little defined the law while operating research on traffic control signals, hence the basis of it as a way to analyze queueing systems.

## **LITTLE’S LAW FORMULA**

Little’s law is incredibly simple.

L = A x W

In this formula, “L” stands for the number of items inside the queueing system. This is also known as WIP, that is, the items that are a “work in progress”

“A” represent the arrival rate and departure rate of items in and out of the business system. This is also known as “throughput” or “the amount of an item passing into and/or out of a system”, and is sometimes represented as lambda. Arrival rate is usually a fraction. This is because you are measuring the rate at which items enter/depart from the system, rather than the number of items or the time between new arrivals. Therefore, A is always expressed as a fraction, that is:

A = (1 item) / (unit of time)

For example, if a new item enters your queue every thirty minutes, then your arrival rate is not thirty, but instead 1/30.

And lastly, “W” is the average amount of time an item spends in your queueing system. This is also known as “lead time” and can also be any unit of time. It’s also known as “lead time” and can also be any unit of time. The confusing part of this element is that the unit needs to be the same as the one you used for “A” that is if you measured the arrival rates in weeks, then “W” will also be measured in weeks.

So, the law becomes:

**Number of items in the system** = (**the rate items enter and leave the system**) x (**the average amount of time items spent in the system**)

The formula can also be changed to make any of the three elements the focus. The three possible variations are therefore:

- L = A x W
- A = L / W
- W = L / A

**A STABLE SYSTEM IS NEEDED TO APPLY LITTLE’S LAW**

Little’s law needs to be applied to a stable system in order to work.

- Unit of measurement should be consistent

This means that you measure the arrival rate in days then the amount of time items spend in the system should also be measured in days.

- Average arrival rate should be equal to average departure rate

This could be a little more difficult to keep consistent. However, the easiest way to deal with this is to alter the arrival rate to match the departure rate. This simply means that do not start any new task until the current one is completed.

- All work enters, is completed and leaves the system

As an extension of having identical arrival and departure rates, your system needs to be one in which work actually leaves it. Having items hang around for an indeterminate amount of time makes both your arrival rate and WIP time completely inaccurate, and so Little’s law can’t be applied.

- WIP amount and time is consistent

If the WIP time is inconsistent. Chances are that you’re trying to apply the law to too much at once. Therefore, need to break your original queue into smaller sub-systems.

For example, rather than examining the entire output of a manufacturing facility at once, narrow your scope to a single type of assembly line or the cycle of a particular product. This way you’re not getting inaccurate results due to the different life cycles of the various products made in the facility.

**APPLYING LITTLE’S LAW FOR ROUGH CALCULATIONS, PREDICTION AND PROCESS ANALYSIS**

The law provides a quick and easy way to perform rough calculations, track performance over time. Also, make predictions for the changes you are planning.

Want to know whether you have the resources to deal with an increase in clients? Use your desired growth rate as your arrival rate, then multiply that by the average amount of time it takes you to deal with a ticket. You’ll then be able to see how many clients will need to be serviced in your support system at any given time. But do not start your business, until you read this blog!- https://explified.com/blog/do-not-start-a-business-until-you-watch-this/

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