Little's Law in a global pandemic

With a pandemic raging and health concerns heightening, businesses and supermarkets are struggling to ensure the safety of their staff and customers. Many retail merchants have been forced to close their doors, and those that remain open must navigate safety guidelines.

Yesterday, as I stood in a big queue outside a supermarket, it got me thinking about queueing theory.

Queueing theory and Little’s Law have been known to the software (and manufacturing) industry for many years. In our world, queueing theory helps us to understand the flow of work through our portfolio of products, projects and services. It can equally be applied to retail shops and supermarkets; indeed, it is often the metaphor used in training materials. 

Littles law is:

L = λW

Or, put another way:

The average number of customers in the store (L) 
EQUALS their rate (λ) of arrival (or departure) 
TIMES the average amount of time (W) they spend in the store. 

How do supermarkets regulate foot traffic?

Let’s put this in terms of a global pandemic, where supermarkets need to enforce social distancing recommendations.

Let’s say you manage a large supermarket, where you need to regulate the number of customers inside the store. You can calculate their average exit rate based on the number of transactions. It could be 5 customers per minute. You can decide your optimal max customers on the premises based on desired maximum crowdedness. For example, 50. Anything lower than a maximum would be fine, but if you go higher at peak times you’re making the system less efficient. It’s better to let people in the queue know the expected wait time (people like predictability, and dislike uncertainty).

In this example, you have 50 customers on your floor at any given time, and five customers leaving every minute. Security outside can let five people in per minute, or adopt a "one out, one in" policy. You could produce a chart for the security team to help to determine expected wait times. For example, if there are 25 customers in the queue, security would be able to tell them their wait time is about five minutes.

If a cashier goes on a lunch break or a self-checkout machine breaks down, and you don’t reduce the number of customers in the supermarket, the system will become sub-optimal, and distancing guidelines will be hard to follow in the shop. You therefore would want to tell security when not all cashiers are available (especially when it is very busy), and they can slow down the arrival rate. 

In this example you’re assuming that the amount of time spent in the supermarket will self regulate. So your formula is this:

50 = 5 (per minute) * W (expressed in minutes)

With this formula you can calculate that customers spend an average of 10 minutes in the supermarket.

Little’s Law in other contexts

Much like the supermarket where there can be pressure from the customers to let more people in to do their shopping, so too in knowledge work (such as software development) there is often pressure from the business customers to increase the number of projects happening. This can happen at an organisation level, program level, or even at a team level.

At the team level in the day-to-day, the temptation is to multitask, but Little's Law tells us that it's more efficient to keep the team focused, reducing the amount of work in progress for an optimal flow through the system.

It can be counterintuitive to limit the number of customers in the shop (or amount of work in progress), but this is what is needed in order to optimise the system and better serve all customers.

A simple formula for a complex system

Little's Law is a simple formula that assumes a stable system, so it is not perfect for variable systems like supermarkets and software portfolios. But it allows us to make sense of our complex environments and helps us to optimise the flow of customers through the shop, or flow of work through the system.

Additional reading