# Little’s Law

Definition of Little’s Law:

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Little’s Law is named after Dr. John C.D. Little, a professor at the MIT Sloan School of Management.  As an expert in Operations Research, he is best known for his proof of the queuing formula L = λW, which is now commonly called Little’s Law.

Little’s Law has been adapted and used as a simple metric for measuring the velocity of a process. This article will discuss the history and derivation of Little’s Law and its adaptation, PLT, as well as the benefits of using it as a measurement of the process.

## Overview: What is Little’s Law?

The original formula of L=λW was developed and published by Philip M. Morse, who challenged his readers to prove that the relationship did not hold for all applications. In words, the long-term average number L of customers in a queue or line is equal to the long-term average effective arrival rate λ of customers multiplied by the average time W that a customer spends in the queue. Little published an article in 1961 showing a proof that confirmed that the relationship is valid for all systems and applications.

Michael George of the George Group and others adapted this concept and applied it to lean manufacturing. The definition then took on two interchangeable names, Little’s Law and Process Lead Time (PLT). While Little had intended his formula to be used in a Queuing Theory context, it turned out that it had perfect application to the concept of Lean. PLT and Little’s Law then became classic metrics for measuring process velocity.

For the application to Lean and PLT, the components of the formula were rearranged so that PLT, or Little’s Law, became PLT=WIP/ER. In words, the Process Lead Time, PLT (the time it takes for an object to go through a process from when it first enters the process until it exits the process) is equal to the WIP or Work in Process (average number of items in the queue or line) divided by the ER or Exit Rate (average number of items that exit the process per a specified unit of time).

To illustrate Little’s Law, let’s use a classic example that is presented in many Lean Six Sigma training classes. Let’s say you took your family to Disney World for a short vacation. Your goal is to select the best rides for the kids so that they spend the least amount of time in line. If we use PLT as your decision measurement, how would that work?

First, as you approach the queue to a ride, say the Jungle Cruise, you would count the number of people already in the line ahead of you, the WIP. You would then count the number of people that leave the line and get on the boat over the course of one minute. That would give you the exit rate or ER.

Now you can do your calculations. Your end goal is to determine the number of minutes you would have to wait in line until you got on a boat. Therefore, the PLT would be equal to the number of people in front of you (WIP), say 160, divided by the ER (Exit Rate) of say 20 people getting on a boat per minute. If PLT=WIP/ER, then the number of minutes you would wait in line would be 160 people/20 people per minute. A simple calculation would conclude that you would only have to wait eight minutes before you and the family got on the boat. Not too bad.

## 3 benefits of Little’s Law

Measuring the velocity or speed of your process is important for managing any process. Some benefits are:

### 1. The ability to reduce the process lead time and its variability

Customers usually want their orders filled as quickly as possible. By measuring and understanding what delays delivery of orders, you can reduce the lead time for your customer’s orders. It will also reduce the variability of lead time so you can better predict when items will be completed.

### 2. The reduction of WIP

By reducing or managing objects in the process, you can reduce the costs of maintaining work in process and free up capital.

### 3. The simplicity of the calculation of Little’s Law.

The formula and computations are very simple. Yet, the results can provide great insight into how the elements of the formula can be improved.

## Why is Little’s Law important to understand?

By understanding how Little’s Law works and the relationship of the factors in the formula, you can analyze and improve any process.

### 1. Know where the fail points are (and how to remove them)

Looking more closely at Exit Rate will allow you to seek out and determine what factors might be slowing the Exit Rate down. Improving those parts of the process can help reduce the process lead time.

### 2. Achieve greater cycles of learning

By reducing process lead time, you get through the production faster and thus more frequently. This allows the process to cycle through more often so that people get better and better — and quality will improve.

### 3. Improved productivity and capacity improvements

Faster exit rates and reduced work in process will allow for greater productivity and potentially increase capacity so that more work can be completed with the same or less resources.

## An industry example of Little’s Law

An online store wanted to improve sales. They developed a social media campaign that they felt would greatly improve traffic to their site. The good news is that it worked. The bad news is that, because of the increased volume, the site kept crashing. Customers were very unhappy because of the delay in accessing the site and the delay in processing their orders and checking out.

The company consulted its Lean Six Sigma Black Belt, Anne, to see if she could help solve the problem. The first thing that she did was complete a simple calculation using Little’s Law. She found that the average number of people in the queue during prime time averaged around 350. The system was able to fully process an order, including searching for the desired products, populating a cart, and checking out and paying at a rate of approximately 80 people every seven minutes, or about 11 people per minute.

Anne then did the calculation of Little’s Law, or PLT, and computed that an average time of waiting to complete a transaction was about half an hour. Of course, customers were not happy with that long of a wait. But, the company did not want to reduce customer demand.

Working backwards, Anne calculated that the Exit Rate needed to be ER=WIP/PLT or ER=350 people/6 minutes per person or 58 people per minute. That was the needed process velocity to satisfy the customer. After some changes in the process coupled with some changes in technology, they were able to achieve this level of service.

## 3 best practices when thinking about Little’s Law

While calculating Little’s Law and PLT is simple, there are still some things to keep in mind.

1. First try to reduce the WIP. By utilizing better inventory and WIP controls, you can reduce the WIP quickly and inexpensively. There is a calculation called WIP Cap that helps you determine the optimal amount of WIP to maintain in your process. The formula for WIP Cap is related to that of PLT.
2. Try reducing Exit Rate by initially looking at the process to see if there is any waste that can be removed. Using the 8 Wastes of Lean as a template, analyze the existing process for any wastes that may be slowing down the exit rate.
3. If all else fails, utilize technology to improve Exit Rate. Spending money on technology should be the last thing you do. Many times, a simple improvement of the process itself will solve the problem. If the current process is already lean and mean and devoid of much waste, then it might be time to invest in more advanced technology to improve the exit rate.

1. What is Little’s Law?

The original Little’s Law is a theorem that calculates the average number of items in a queue or line, based on the average waiting time of an item and the average number of items arriving at the system per unit of time.

1. What is PLT?

PLT, or Process Lead Time, is an adaptation of the calculation for the original Little’s Law that is used to compute how long it will take an item to be completed once it enters a process queue.

1. How do you calculate PLT and Little’s Law?

PLT is equal to the WIP (number of items in the process or work in process) divided by the ER (exit rate or how many items leave the process per unit of time). The resulting computation is a measure of time.

## A little summary of Little’s Law

Little’s Law, also commonly referred to as Process Lead Time (PLT), is a powerful metric to measure the speed and throughput of any process. The PLT is a function of the number of items already in the process queue (WIP) and the speed at which items leave the process (Exit Rate). The relationship of these factors is PLT= WIP/ER.

You can use this concept and calculation to determine something as simple as what line you should get in at the grocery store in order to leave as quickly as possible. You can also use it for something more complex, such as patient scheduling for the operating room at a hospital.

Next time you’re in line at the company cafeteria, calculate how long it will take you to get to the cash register and pay so you know how long you have to eat before your lunch break will be over.

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