Little’s Law is a simple but powerful relationship that operators, managers, engineers, and management consultants must understand. Applications of Little’s Law are widespread. MBAs will encounter Little’s Law in their first operations course in an MBA program. Our operations tutors can help you understand what Little’s Law is and how to apply Little’s Law in a variety of settings. On this page, we cover the following Little’s Law topics:
- Introduction to Little’s Law
- What is Little’s Law?
- Applications of Little’s Law – Service Businesses
- Applications of Little’s Law – Operations Management
- Little’s Law Vs. Queuing Systems
- Assumptions of Little’s Law
- Common Errors in Applying Little’s Law
- Little’s Law Deals with Averages
- Little’s Law: A Little History
Note: Little’s Law is distinct from Littlefield Technologies. We provide tutoring for Littlefield Technologies here.
Introduction to Little’s Law
Waiting time and the length of a queue are important customer service drivers for a product or service. So it is natural that CEOs, managers, owners, and operators are very interested in the average waiting time and the average number of people in the queue or waiting for service are variables. Little’s Law helps you understand the relationship between these metrics. Little’s Law has widespread applications in the world of business and management.
Some operations courses that we tutored Little’s Law for include EMBA Tutoring for B6102 Operations Management, B5102 Operations Management & B8107 Service Operations Management at Columbia Business School, MBA Tutoring for Operations Management & Research at Chicago Booth School of Business, Tutoring for ISOM 550 – Data & Decision Analytics Course at Emory, etc.
What is Little’s Law?
Little’s Law says that the average number of people waiting in a queue is equal to the arrival rate multiplied by the average time to provide the service.
The arrival rate is the average rate at which items arrive in an operation, system, or business. The average time is the time that a customer spends in the system. Professors and practitioners use a variety of annotations, but if we use
- L =average number of people waiting in line,
- W = average waiting time for each person in the system, and
- A =average number of customers entering the business
Little’s Law says
- L = A * W
This relationship, outlined by Little’s Law, is remarkably simple but incredibly useful.
Applications of Little’s Law – Service Businesses
Little’s Law has numerous applications in operations management, business, and managerial decision-making. A classic example is a business with a queue to which customers arrive, spend some time getting service or work done, and then leave. Little’s Law helps you estimate queue length from waiting time and vice versa, or more specifically, estimate arrival rates from queue length and waiting times and vice versa. This is especially useful when one or two of these inputs are easy to measure and the others are not.
Applications of Little’s Law – Operations Management
An operations manager can use the relationships defined by Little’s Law from another perspective. An operations manager is examining cycle time, work-in-progress (WIP) inventory, and throughput to address operational management challenges. So instead of L, W and A, the operations manager uses cycle time, work-in-progress (WIP) inventory, and throughput. The Little’s law relationship still holds, albeit with different annotations.
- W = WIP = L =average number of people waiting in line,
- T = Cycle Time = W = average waiting time in the system each person, and
- R = Throughput = A =average number of customers entering the business
Little’s Law Vs. Queuing Systems
Students often ask what is the difference between Littles Law and Queuing theory. Queuing theory deals with a very broad framework. There are many queuing systems and models. Each queuing system has a different model for example G/A/1, G/G/1, etc.
Little’s Law deals with ONLY one specific model of queuing systems within the larger field of queuing theory. Little’s Law provides a simple way to relate cycle time, WIP, and throughput or L, W and A as applicable. Queuing theory is a more comprehensive framework for analyzing and optimizing queuing systems.
Let’s look at some applications of Little’s Law.
Ageing
Mark loves aged wine. However, given the cost of aged wine, he usually purchases only relatively recent vintages. He wondered how long his wines had aged in his possession.
Mark estimated that he has about 120 bottles of wine on average in his wine cellar. Consumption depended on several factors, including the number of guests each week, the season, and work-related travel demands. However, Mark estimated that the consumption was about two bottles a week. Can you help Mark estimate how long his wines aged in his possession?
Lead Times
Orders are received by a welding service at a rate of 15 orders per week. Twenty welders are working on the floor. Each welder is assigned to only one order at a time and achieves a 90% utilization rate throughout the year. How much lead time should you plan when placing orders?
Processing Times
Joe gets 10 emails per hour during working hours. If she has 140 unopened (not responded) emails, how long does it take Joe to answer her emails?
Casualty Ward
The emergency department has 20 beds, which are 60% occupied on average. Patients stay in the emergency ward for 36 hours. How many patients does the emergency department get per day?
The Port Authority
Ships arrive at the port authority at the rate of 60 ships per week. Due to the recent strike, there are additional staff to help clear backlogs. If there are 300 ships in the port, how many days does it take to clear a ship?
Sam’s Used Cars
Sam estimates that he has 42 cars in inventory on average. If he can get 15 used cars per week to sell, how long does it take to sell a car on average?
NY Housing Market
The NY housing board believes that, on average, 2500 new apartments come to the market each day. On average, it takes 9 months to sell an apartment. How many apartments are available for sale in NYC?
Assumptions of Little’s Law
Little’s Law operates under some simplifications and assumptions. Little’s Law underlying assumptions are:
- Steady state assumption: Little’s Law assumes that the operation must be in a steady state. By operation must be in a steady state, we mean that the arrival and departure rates must be relatively stable.
- Input = output assumption: Little’s Law assumes that all people entering the business will be served and depart over the considered time frame. In other words, all work entering the system must be completed and sold. This can also be viewed as a no loss due to pilferage, abandoned items, evaporation, etc., and
- Average que length: Little’s Law assumes that the average number of people waiting, or inventory, or work-in-progress (WIP) is relatively constant. The system load is not increasing nor decreasing during the time frame considered. This is termed as “end effects,” referring to the impact of the beginning inventory and ending inventory on the system or business.
Common Errors in Applying Little’s Law
As MBA tutors, the most common error we see among business school students is identifying settings to apply Little’s Law. Students get stuck, not knowing that Little’s Law can be applied to a particular situation or operations question. Only when prompted do students proceed to identify cycle time, WIP, and throughput or L, W, and A as applicable..
The second most common error we see as operations tutors is mixing up units for Little’s Law ingredients L, W, and A. Consistent units of measurement for cycle time, WIP, and throughput, or L, W, and A as applicable. Getting the right units is crucial; otherwise, you will end up with wrong answers.
In operations management, the throughputs are expressed in terms of output rates rather than arrival rates. This stumps a student looking only for arrival rates, so beware how case studies or questions provide inputs.
Sadistic professors may provide interarrival times instead of arrival rates. Students in a hurry end up mixing up the arrival rate with interarrival times. So, remember to convert interarrival times into arrival rates to use as an input into Little’s Law equation.
Students often misunderstand the inputs, as the question may use different words for the inputs. For example, average service time may be described as flow time, throughput time, cycle time, etc.
Little’s Law Deals with Averages
Little’s Law deals with averages! When using Little’s Law, students need to keep in mind that Little’s Law works when there is sufficient length of time considered. For example, if you want to understand the average length of time a customer waits for a coffee at a drive-through, you must consider that the traffic will be higher during the rush hours and lower otherwise. However, if you consider a 24-hour day, the averages will work. You can also take a subset, and the Little’s Law relationship should work, provided the “end effects” do not come into play (beginning inventory and ending inventory are equal).
Little’s Law: A Little History
Little’s Law is named after John D.C. Little, a professor at MIT. However, Little’s Law was not discovered by Professor John D.C. Little.
Little’s Law was first mentioned in passing by Cobham in 1954 in an article on priority queues. Cobham A stated, without providing any proof, in the article titled ‘Priority assignment in waiting line problems’ (Oper Res 2(1):70–76) that the expected number of units of priority k waiting to be serviced is the arrival rate A times the wait time W.
The explicit formula was first stated by Philip Morse in his book, ‘Queues, Inventories and Maintenance’ (Morse, 1958). Philip Morse does not name the relationship, but clearly states the relationship in the Little’s Law equation for the first time.
It was Professor John D.C. Little who first provided proof of this relationship in 1961 and published a paper with the title “A Proof of the Queuing Formula: L =AW,” including the now-famous buildup diagrams. This led to the relationship being named after Professor John D.C. Little as Little’s Law. Professor John D.C. Little’s most important finding was that Little’s Law holds in a queuing system when observed over a sufficiently long period.
Since then, many additional papers have been published, including Jewell (1967), Eilon (1969), Kleinrock (1975, 1976), Stidham (1974 & 2002), etc.
Operations Research Tutoring
Our operations research tutors can assist you with tutoring to understand and apply Little’s Law in service and manufacturing operations. Other operations topics we can assist you include queuing theory and waiting lines, decision trees, linear programing using Microsoft Excel’s Solver, newsvendor models, batch processing, Littlefield simulation games. etc. Feel free to call or email if we can be of assistance with live one on one tutoring.