No matter what type of work you do, it helps to have frameworks to guide your decision making. We aren’t programmed with solutions to several problems we face; many times it seems we must “go with our gut” or gamble with decisions – but is that true? Having something more empirical and concrete behind our decisions makes us more rational, and I argue, more confident. Thankfully, over the past thousands of years, several other humans have made some pretty cool discoveries in the area of mathematics and its application to real-life conundrums. Some simple formulas will give you an edge when it comes time to solve certain problems regardless of whether you are a business practitioner or a programmer, hands-on or strategic, senior or just entering the workforce. For Little’s law, you don’t need to have an MBA or mathematics background.
Our first magic formula is Little’s law. This theorem is not thousands of years old but was originally described and proven by John Little in the groovy early 1960s. To use it, the only pre-requisite is basic algebra knowledge.
Problem: I need to know the lead time, current workload or flow rate of a system.
Solution: Little’s Law
This robust little formula can be expressed as :
L = λ x W
where:
- L = Work-in-progress (WIP), a.k.a. the average number of items in a queuing system
- λ = Throughput, a.k.a. the average number of items arriving or departing per unit of time (rate per minute, hour, day, etc.)
- Note: You will find that throughput is either identified by a lowercase lambda (λ) or by a capital A across different resources. I used a lambda so I appear like a mathematical whiz. 🧙♀️
- W = Lead time, a.k.a. the average waiting time an item spends in a queuing system
Little’s law has gained much of its prominence in the project management world; it is relevant in decision making in regards to operations and managerial finance. The basic use of this formula is to analyze queuing systems, that is, systems that have pieces of work that enter (input) and leave (output). It is ideal for solving capacity, staffing or flow issues.
Little’s law must be applied in a steady-state system, meaning inputs (raw materials, or queued customers) turn into outputs (finished goods, or serviced customers), without anything lost in the middle. Think restaurants, medical practices, production lines, kanban cards. It is very likely that there are multiple stable systems in one business.
Simplifying Little’s law as much as possible, it can be used to calculate WIP, throughput and lead time as long as at least two of those pieces are known.
A guided example: Walk-in dog grooming business
Let’s say we have a busy, walk-in dog grooming business. We want to better manage customer expectations by more accurately understanding waiting times. Our business has been in operation for a couple of months, so we have an idea of how long it takes for us to groom dogs as well as usual demand in the waiting room.
- L – Work-in-progress (WIP) = On average, we have 6 dogs waiting patiently to be groomed.
- λ – Throughput = On average, 7 dogs are called in for grooming every 2 hours. We can simplify that into an hourly rate of 3.5 dogs/hour.
- W – Lead time = ????? – Since we know WIP and throughput, we can solve for lead time, which is in this case, the customer waiting time.
If we rearrange the formula shown earlier, we can solve for lead time using this configuration:
W = L / λ
Let’s plug in the throughput and WIP values we have established.
- W – Lead time = 6 dogs / (3.5 dogs / 1 hour)
- W – Lead time = 1.71 hours
Therefore, front desk staff and customers can expect a waiting time of 1.71 hours based on the average WIP and throughput. Having a clear idea of waiting time can help managers plan for staffing, budgeting and caring for customers in general. Perhaps complimentary beverages, dog toys or other amenities could reduce the pain of a nearly 2 hour waiting time.
Alternatively, let’s say that such a waiting time seems way too long for customers to expect to wait. While we do offer the convenience of walk-in appointments, we want wait times to be about 1 hour. We could manipulate our work-in-progress (waiting customers) or throughput (rate of serving customers) to bring lead times down in the waiting room. Obvious ways to do this would be to accept fewer customers once the waiting room exceeds a certain capacity or to hire more dog groomers to increase the number of dogs going through the system.
Other possible applications of Little’s law
The universality of Little’s law is what makes it so valuable and applicable in real-world business. Here are a couple more examples of where it could and should play a role.
- Software development workflows
- Response times of support teams
- Emergency room staffing
- Restaurant seating capacity
Using Little’s law
Now that you are well-acquainted with Little’s law, try to apply it to a stable system you use at work or home. Little’s law doesn’t give you the right answer if you put in some actual information about a system. It merely gives you the reality. However, with your understanding of Little’s law, all it takes is manipulation of the “levers” in the equation to bring one of the target factors to your desired level. Increasing throughput will definitely reduce lead times. If you cannot afford the machinery or humanpower to increase your throughput, then you could accept fewer jobs, as less WIP will also reduce lead times. However, in this case, you will lose some business.
It’s all about making the mix of WIP, throughput and lead times fit your needs. It will be up to you or the business to decide what constitutes an appropriate lead time, WIP and throughput, but Little’s law goes a long way to analyzing and understanding queuing systems. Hopefully, it’s something you feel you can add to your repertoire of systems knowledge.