Little’s Law

One of the simplest yet most powerful theorems known to man is called Little’s law. This law can be seen in our daily lives. You might not be aware of it, but you can be partaking in an activity where Little’s law can be observed. Wherever there’s a queue, Little’s law is put into action. We queue in hospitals, train stations, groceries, service shops, restaurants, banks and many more. 

But queues are not only experienced by humans. If we treat each process as a system, we can see aspects where queuing is observed. Manufacturing plants and factories have many queuing systems within their operations. Paperwork and documents can symbolize a queue in an office. Even tasks that make up our projects at work are also queued. Little’s law can apply to any scenario where an item or person waits to be processed or serviced.  

What is Little’s Law?

In 1954, a renowned Massachusetts Institute of Technology (MIT) professor named John Little developed this theorem we call Little’s Law. But it was only in 1961 that he formally published proof of his theorem and concluded that the law holds for all queuing scenarios. Little’s law became a fundamental concept in queuing theory and the study of operations research.

Little’s Law Formula

Little’s law states that the average number of items within a system is equal to the average arrival rate of items into and out of the system multiplied by the average amount of time an item spends in the system. Sounds a lot to take in. So, let’s try and simplify things.

Little’s Law Simplified

Little’s law applies to anything that has a queue. To illustrate Little’s law in a simplified manner, let’s use an example which almost all of us will be familiar with – the coffee shop. To make things even simpler, we’ll imagine this coffee shop is a grab-and-go establishment. No table service is provided.

In our formula above, “L” represents the number of items inside the system. So in our coffee shop example, these will be the customers that come to the store for their usual coffee fix. This variable is also called WIP or Work-In-Progress items. 

Next is “λ” (lambda) or the arrival rate of items within the queuing system. This one can be a little tricky. But let’s go back to our coffee shop example to visualize things. As in all queuing systems, the goal is not to stay in the queue. True to our example, customers will line up for their coffee and will leave the store as soon as they get their order. In this sense, the rate at which customers pass into and out of the system is what “λ” represents. Take note that this variable is denoted per unit of time. In our coffee shop example, let’s say they get 1 customer per 5 minutes. This means that the system’s λ is ⅕ or 0.2. This variable is also called throughput. 

Lastly, we have “W” or the average time an item spends inside the queuing system. In our example, this represents how long a customer waits to get their coffee. Note that the unit of time used in “λ” should be the same as what you’ll use for “W.” This variable is also called lead time.

In our coffee shop example, let’s say it takes 10 minutes from the time a customer places an order until they receive their order. Given that our throughput is 0.2, this means that our WIP is 2. At any given time, you’d see an average of 2 customers in the store waiting to get their piping hot coffee.

Little’s Law and Kanban

Many principles of methodologies like Kanban and Lean take their cue from Little’s law. One of the core properties of Kanban is to limit Work-in-Progress (WIP). This is precisely because it allows for a predictable Kanban system where throughput can be made accurate at any given time. This core principle allows any Kanban system to have a well-managed and smooth flow. 

In Kanban, Little’s law can be expressed in this way using the alternative terms we discussed in the previous section:

And you can easily manipulate the formula to focus on any of the variables depending on what you need to measure. 

It is important to note though that for Little’s law to hold, the system being observed must be in a steady-state condition and the units of measure for all three variables are consistent. Achieving a steady-state condition means that on average, the arrival and departure rate of items into and out of the system remain consistent. 

Does this mean that it has to be the same every single time? Not necessarily. As we’re dealing with averages, these assumptions can afford not to hold for the entire duration of the period being observed. But do note that as you deviate more from the average, the lesser the accuracy of the equation will be. Therefore, it’s important to examine the system before using Little’s law and ensure the assumptions can hold for the majority of the time. Otherwise, insights derived from it may be inaccurate and unreliable. This is where limiting WIP proves to be helpful for systems where the arrival and departure rate of items fluctuate. 

The beauty of Little’s law is in the simplicity of it. It can be applied to a myriad of systems and processes and can help businesses grow and improve. Once you embrace Little’s law, you’ll see systems with a different lens and can spot opportunities to improve its overall performance.