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Little's Law (Explained With Tacos)

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Little's Law (Explained With Tacos)

What is Little’s Law?

The power of this formula lies not just in the simplicity of what it does, but in what it doesn’t do.

A simple definition:

Little’s Law states that the long-term average number of customers in a stable system L is equal to the long-term average effective arrival rate, λ, multiplied by the average time a customer spends in the system, W.

Expressed algebraically, Little’s law appears quite simple: L = λ W Little's Law Formula

L represents a business’ average number of customers.

λ refers to these customers’ arrival rate (assuming an identical departure rate).

W symbolizes the average amount of time customers spend at the business.

When calculating L, the number of people at a business, simply invert the equation: λ W = L

Little's Law Formula when calculating the number of customers

Imagine 20 customers visit your hip and trendy taco truck every hour (λ = 10). They stick around for half an hour (W = .5), eating and chatting with each other. At any given time, you would have an average of 10 customers standing around your food truck and enjoying your organic, grass-fed shitake mushroom and carne asada burritos:

λ W = L

20 x .5 = 10

Now, imagine you hired a team of acrobatic sign-spinners to promote your business and this effort doubled your arrival rate to 40/hour. You would have to chase people away from your truck with squirt guns to make room for new customers:

40 x .25 = 10

…or cut a second serving window in the side of your truck to serve two lines of customers:

40 x .5 = 20

Now that you’ve wrapped your head around the basics of Little’s Law (and worked up an appetite), you can appreciate how corporate leaders use this little formula to manage workflow and increase efficiency.

Little's Law Formula Throughput

Why is Little’s Law Important for Businesses?

Despite its simplicity, business leaders gain powerful insights from Little’s Law. However, they use different terms when working with this formula because they’re calculating the capacity of systems, not retail outlets:

L =- Work in Progress (WIP) instead of customer volume.

λ =- Throughput (departure rate or production output) instead of customers’ arrival rate.

W =- Lead time (the time an item spends in a system) instead of the time people spend at a business.

So, L = λ W becomes…

WIP = Throughput x Lead Time

Little's Law Formula WIP

Managers flip this equation around when calculating Lead Time:

WIP/Throughput = Lead Time

Little's Law Formula Lead Time

Imagine a company that manufactures stylish, hand-crafted beardnets for taco truck employees at the rate of 10 a day (Throughput). Because Little’s Law assumes equal arrival/departure rates, the manager of this business can use the number of beardnets departing the system as the arrival rate (λ=10).

By looking around the shop, this manager can see 5 employees, each diligently and lovingly weaving a beardnet from non-GMO, fairly-traded Chilean yak tail hair. Given this number of employees, this production team averages a WIP of 5 (W=5).

WIP/Throughput = Lead Time

5 beardnets at a time/10 beardnets a day = .5 day to produce one beardnet

Toggl helps you quickly determine your team’s workflow and identify areas of low/high efficiency. Our quick and simple time management tools allow you to easily organize your timesheet data by tags or projects. With this essential information, you can quickly calculate your WIPs, Throughputs, and Lead Times and make real-time decisions to increase your organization’s efficiency and productivity.

Little’s Law vs. Kanban

Despite the common misconception expressed in the heading above, Little’s Law and Kanban aren’t at odds with each other. In fact, the creators of popular management strategies like Kanban, Lean Manufacturing, and Agile based many of their foundational principles on Little’s Law.

Kanban software development team leaders measure WIP in terms of scenarios, user stories, cards, etc. Because arrival/departure rates are equal in stable systems, they can use their Throughput as the Little’s Law arrival rate. With this information, they can calculate the amount of time their team needs to complete a task. Managers use this formula not just to determine the “engineering lead time” of projects, but also “order lead time” (the communication delay time between a customer’s order and the beginning of production).

Kanban managers use Little’s law for much more than predicting lead times, WIP, and throughputs. They use this formula to give their team members a broader perspective on workflow and efficiency.

Imagine our beardnet weavers wanted to double their production (Throughput) to meet a spike in the demand for stylish and sanitary culinary face wear. Flipping around the equation, their manager notices that:

WIP/Lead Time = Throughput

Little's Law Formula Throughput

To double the company’s Throughput, this manager could hire 5 more employees (doubling the team’s WIP):

10 beardnets at a time/ ½ days to produce one beardnet = 20 beardnets per day

Alternatively, the original team of 5 weavers could work twice as fast (by adopting a new breakthrough in beardnet weaving technology) and cut their lead times in half:

5 beardnets at a time/ ¼ day to produce one beardnet = 20 beardnets

Little’s Law vs. Cycle Time

Many people confuse Cycle Time and Lead Time, perhaps because they both involve “time.” You can easily make sense of these concepts by realizing that lead time measures elapsed time and cycle time involves time per unit. If you find yourself adding or subtracting cycle and lead time from/to each other, stop. The units don’t match, and you’re heading for trouble.

Cycle time is the average time it takes to complete one step (or many steps) of an operation.

Imagine our taco truck offers freshly-made tortilla chips along with a tangy salsa verde. These taco-slingers have two tortilla presses, each of which flattens one tortilla per minute. The cycle time of one press is 1 minute per tortilla; the cycle time of this company’s fleet of two presses is 30 seconds per tortilla. Their deep fryer is just large enough to handle 4 tortillas every two minutes (cut into wedge-shaped chips, of course) so it’s no problem to churn out the chips.

If this truck’s regular customers re-discover their passion for the first Ghostbusters movie and all foods that resemble green slime, their demand for salsa verde will dramatically increase. If this company buys two more tortilla presses, they can now make tortillas twice as fast (15 seconds per tortilla).

However, this action would create a bottleneck. Though the physical capacity of the fryer is adequate, its cycle time is not. At 2 minutes for every 4 tortillas, its cycle time equals 30 seconds per tortilla.

This company needs to invest in a larger deep-fryer, too!

In our taco truck example, the lead time of a tortilla chip (before any investments in new/additional equipment) equaled 3 minutes: 1 minute in the press and 2 minutes in the fryer.

However, adding the cycle times of the presses (30 seconds) and the fryer (30 seconds) doesn’t add up to the lead time. Only when we consider that this company operates two presses do we see that this system was running at full capacity before their regulars went crazy for green, slimy foods.

Now that you understand cycle time a little better, reconsider its difference from lead time.

In Little’s Law, Lead Time (W) refers to the average amount of time an item spends in a system; Cycle Time measures Throughput (λ). L refers to the volume of items in the system.

L = λ W

In the tortilla chip example,

WIP (3) = Throughput (1) x Lead Time (3)

This system can hold three tortillas at a time (WIP), taking into consideration both the physical and temporal capacities of the presses and fryer. Yum!

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Practical Implications of Little’s Law

John Little and his colleagues have written extensive proofs of Little’s Law, which later researchers have used to expand the application of this formula to many facets of business. In a nutshell, you can use Little’s Law to examine the relationships between the speed that items that enter and exit a system (λ or Throughput), the capacity of a system (L or WIP), and the time they spend “waiting” in-system (W or Lead time).

The power of this formula lies not just in the simplicity of what it does, but in what it doesn’t do. When using Little’s Law, you don’t have to create a complicated model of your operations by taking into account the size of your team and whether items come to them via one queue or many. You can disregard service and inter-arrival time distributions – and even the order in which your team works on items.

Who Is John Little?

John Little, chair of Management Science at the MIT Sloan School of Management, has worked in the fields of marketing and operations methodology for over 50 years. In addition to creating and proving Little’s Law, he has made significant contributions to the fields of traffic signal systems, decision-support, and digital marketing.

Little and his colleagues began developing his Law as early as the 1950s. This formula, Little’s most well-known contribution to the field of business management, became a cornerstone of modern organizational theory. Researchers continue to expand on his theories and unravel the intricacies of pre- and post-system behaviors, wait times, and time probabilities.

At the advent of the Internet, Little quickly adopted this new technology and taught Sloan’s first internet marketing course. He continues to research marketing automation for internet retailers and the flow of packaged consumer goods. In addition to teaching at Case Western and MIT, he founded the Kana Software and Management Decisions Systems companies.

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