Many of us have ordered from Domino’s or McDonald’s and noticed that the average wait time is under 30 minutes – a testament to their operations management and efficiency. This consistent performance, irrespective of the millions of orders, is partly due to a mathematical theorem known as Little’s Law. Have you ever wondered how they are able to achieve this efficiency?
What is Little’s law?
John Little, an MIT professor defined Little’s law, which says that in a steady state queuing system, the average number of items i equals the average arrival rate multiplied by the average time that an item spends in the system.
L = λ x W … Equation (1)
- L = average number of items in the system
- λ = average number of items arriving per unit time
- W = average waiting time in the system for an item
Little’s Law and Kanban: Optimizing E-Commerce Workflo
Imagine an e-commerce website that receives an average of 500 customer visits per hour. Each customer spends approximately 15 minutes (0.25 hours) on the site. By applying Little’s Law — L = λ × W, where λ is the arrival rate and W is the average time each visitor spends on the site – we calculate that the average number of customers on the site at any given time is 125.
- λ = 500 customers per hour
- W = 0.25 hour
- L = Avg. number of customers = 500 * 0.25 = 125
This statistical insight into the site’s load is crucial for effective operational planning. Knowing the average percentage of consumers who convert into buyers allows the company to implement a Kanban system in tandem to effectively manage orders, significantly enhancing the efficiency and application of handling this workload.
The Kanban system visualizes and optimizes the workflow necessary to manage customer interactions smoothly. It ensures that essential tasks such as order processing, customer service, and inventory updates are completed efficiently without overburdening any part of the system. Additionally, the system aids in anticipating the expected workload and maintaining adequate supplies, which are essential for promptly meeting customer demands and ensuring sustained service quality.
But now back to Little’s Law: Suppose that the website added features based on virtual and augmented reality to have a better online shopping experience for its customers. This resulted in an increased length of time that a customer stays on the site. The website observed that the average time spent by the customers increased to 30 mins (0.5 hours). Applying Little’s law, we
can infer that the average number of customers at any time visiting the website became 250. Someone at the company observed that the website needed to add infrastructure to support 250 concurrent customers.
- λ = 500 customers per hour
- W = 0.5 hour
- L = Avg. number of customers = 500 * 0.5 = 250
Taking the example further, consider that the website added a new social media campaign that increased the customer traffic to 1000 customers per hour. Applying Little’s law, we can infer the average number of customers at any time visiting the website became 500. Someone at the company observed that the website needed even more infrastructure to support this.
- λ = 1000 customers per hour
- W = 0.5 hour
- L = Avg. number of customers = 1000 * 0.5 = 500
The beauty of this deceptively simple-looking law lies in the fact that the law holds true irrespective of various other factors such as distribution of service time and arrival time, number of input queues, the order in which input requests are processed, and more.
Little’s law is a very simple equation, but many things can be inferred from it. In all its simplicity, this law has found its application in many industries such as manufacturing, software development, and resource staffing, among others. Let’s see how different things can be inferred and the impact of this law in various use cases.
Little’s law in search engines
Search engines came into existence in the 1990s and changed the way we discover information, conduct research, shop for products and services, and connect with others. Search engines now process over 99,000 queries every second, making over 8.5 billion search queries a day. From the beginning when search engines were solely based on text queries, we now have search engines that begin search from voice and visual input as well. Within 30 years of the development of the first search engine, the search engine market is already generating over 200 billion USD in revenue every year. The numbers clearly indicate how the arrival rate (λ) grew across the years and from the revenue we can infer how successful the search engines are.
Applying Little’s law to this space:
Number of users in the system (L) = Arrival rate of search engine requests (λ) * Time spent by users on the search engine (W).
A very high arrival rate carries the risk of having a very large number of users (L) being present in the system. A very large number of users in the system either carries the risk of high wait times or demands a lot of infrastructure.
However, while search engines see a very high arrival rate, the users do not stay on the site for too long. They quickly move on to some other site or exit. Hence, the companies that offer search engines primarily focus on improving the time spent by the users. This is done using various levers such as improving search results, making them context-aware, and improving the user experience to quickly find and help users navigate to the information of interest. This not only improves the user experience, but also significantly reduces the W (avg wait time) and in turn the L (number of users in the system).
Little’s law in retail stores
Retail stores present another interesting application of Little’s law. On average, a typical person spends one year, two weeks and a day in a retail store queue in his lifetime. The research also found that after nine minutes, shoppers are likely to give up queuing and leave empty handed. 86% of consumers will avoid a store if they think that the queue is too long. If the company loses a lot of customers, the arrival rate goes down sharply.
Applying the Little’s law to this space:
Number of customers in the store (L) = Arrival rate of customers in the store (λ) * Time spent by customers waiting in the queue and in the store(W).
Whenever the number of customers in the system is greater than the number of customers a system can accommodate, then the additional customers are supposed to wait. This increases the amount of time spent, causing some customers to exit the queue and leave before they make a purchase. Hence, companies focus on reducing the time spent by the customers in a queue.
Companies take various measures to reduce the time customers spend waiting. Some retail stores offer automated retail experiences where there are no cashiers, no payment is accepted at the time for purchases, and there are no queues. A customer enters the store, picks up their items and then exits. The store uses several technologies including computer vision, deep learning algorithms and IoT for the purchase, payment, and checkout of a retail transaction. On an average, on high demand days, the waiting time in a traditional retail system is two hours and six minutes while in an automated retail store, the waiting time is around one hour. Some of the companies using automated retails stores are Amazon Go, Alibaba’s Hema store, Jack and Jones, and Vero Moda.
Little’s law in manufacturing: Reducing Cycle Time
In manufacturing, where metrics like cycle time are crucial, Little’s Law guides decisions. Work-in-progress aligns with the average throughput and lead time—the average amount of time each unit spends in production. Little’s law translates to manufacturing industries as follows:
- Number of customers in the system maps to the work in progress.
- Customer average arrival rate maps to the throughput, i.e., the production output.
- Average time a customer spends in the system maps to lead time, i.e., the time an item spends in the system for a machine to act on.
Little’s law here looks like the following:
Work in progress (L) = Throughput (λ) * Lead time (W)
Industries need to increase the throughput (λ) i.e., the supply of their product to meet the increased demand over time. For this, they either increase the capacity of their production unit (L) or decrease the lead time (W). Increasing the production capacity comes with certain limitations – setting up more manufacturing units involves huge capital investments, which can cause saturation in the production capacity over time. Hence, industries mostly focus on decreasing the production time by enhancing, automating, and upgrading their manufacturing units with the latest technology.
In recent times, automation has seen a tremendous growth in the manufacturing sector. The research shows that automation would create a significant impact in this sector. According to a research report, the world economy might gain $4.9 trillion annually from manufacturing automation by 2030. Automating 64 percent of manufacturing tasks can save 749 billion working hours. Lot of electric, automobile industries are already highly automated to decrease the production time and increase their supply. Tesla automated most of their manufacturing plants and already reached a potential to produce 2 million electric cars annually in the year 2022.
Agile Project Management: Enhancing Efficiency with Little's Law
Little’s law has its application in agile methodology as well, enhancing project management and predictability. Software development team leaders actively use this law to estimate the status of their deadlines, efficiency, and performance of their workflow. To better understand the relationship of lead time in lean practices, we rewrite Little’s law as follows:
Lead Time (W) = Work in progress (L) / Throughput (λ)
The law then translates to agile methodology as follows:
- Work in progress maps to the number of user stories picked for development.
- Throughput maps to the number of user stories that the team can implement in a unit of time (week).
- Lead time maps to the time the team will take complete its backlog.
Little’s law this looks like the following:
Time to complete backlog (weeks) = Total number of user stories picked / Number of user stories that the team can finish in a week.
The focus of the software team managers is to improve the throughput, which reduces the time taken to deliver a project. This can be done by either improving the performance of the team members, decreasing the lead time (W), or adding more people to the team, increasing the work in progress (L).
For example, in a project there is a requirement to develop 100 new features to the existing product. With the current team, they can develop 5 (L) features of a project at a time where each feature takes 0.5 days to develop. Applying the Little’s law, the current throughput is 10 features per day, making it 10 days to complete the development of all the features.
- L = 5 features at a time
- W = 0.5 days
- λ = 5/0.5 = 10 features per day
Say, the team size is doubled to increase the number of features developed at a time making it 10 (L) features developed at a time. Applying Little’s law, the updated throughput is 20 features per day, which means it takes 5 days to complete the development of the 100 features.
- L = 10 features at a time
- W = 0.5 days
- λ = 10/0.5 = 20 features per day
Instead of increasing the team size, the team receives training that helps them reduce the time to develop a new feature by half. Applying Little’s law, the updated throughput is 20 features per day, which means it takes 5 days to complete the development of all the features.
- L = 5 features at a time
- W = 0.25 days
- λ = 10/0.25 = 20 features per day
Note that in addition to using Little’s law to estimate and plan process performance, its real benefits are leveraged in understanding the bottlenecks that prevent reduction in the lead time or increase in the throughput. These inefficiencies can then be addressed using various measures such as streamlining processes, ensuring sufficient capacity, improving team composition, and more.
Conclusion
Little’s law provides the organizations with three main benefits – decision making, identifying problems, and identifying areas of improvement. Despite the simplicity of the law, the law can be observed in diverse domains solving complex problems across various levels of the company from small-scale startups to large enterprises and government bodies.
