Objectives

Upon completion of this lesson, you will be able to:

  • state Little’s Law
  • use Little’s Law to calculate the execution time of a process using two empirical observations

Motivation

Little’s Law is an important law for calculating various process characteristics that comes from queuing theory and operations research. It applies to any environment where requests for processing arrive at some system for processing and then are processed at a certain rate.

John Little, a former professor at the Massachusetts Institute of Technology (MIT), discovered this theorem in 1954. He first applied the theorem to queues in shops as part of his work in operations research (what is now generally called quantitative analysis). Later, it was found that it to anything that can queue – including business processes, manufacturing processes, and even software development processes.

The law describes the relationship between a unit of production and the process that produces it. For example, a unit of production can be a widget (manufacturing), an outcome of business value (business process), an information object (system process), or an implemented requirement (software development process).

Little’s Law

Little’s Law states that the long-term average number (I) of units of production in a process (viewed as a stationary system) is equal to the product of the long-term average effective throughput rate (λ) and the average time a unit of production takes to go through the process (T).

\[ I=\lambda\times T \]

Flow-Time

We can use Little’s Law to calculate the average flow time T of a process, i.e., the average amount of time that it takes to go through the process for a single unit of production.

\[ T = \frac{I}{\lambda} \]

This would give us the execution time of a process without having to find the execution time of individual activities, calculate the average time for divergent paths, or estimate frequencies with which paths are taken. We only need two variables: the average number of units waiting or being in some state of processing (i.e., the inventory I) and the rate at which units are produced (i.e., how many units are leaving the process during some period of time). Both of these can be observed and counted – of course, we would want to count several times at different times, i.e., take a sample, and then average those out.

Example I

For example, for an airline passenger check-in process, the unit of production would be a “checked-in passenger” and therefore T would be the average amount of time it takes to check in a passenger. Let’s say that we do several observations at different points in time and on different days and we find that on average there are 18 passengers waiting to be checked in or are being checked in by one of several agents and that during a 20 minute period, 11 passengers finish their check-in. So, we would have I = 18 passengers and λ = 33 passengers per hour (11 per 20 minutes, so 33 per hour).

Putting this into Little’s Law, would give us

\[ T = \frac{I}{\lambda} = \frac{18}{33} = 0.54 hrs = 32 mins \]

So, the average passenger takes 32 minutes to check in – quite slow. Unfortunately, we do not know where the bottleneck is; that would require a time-study and a visualization of the process.

Example II

A fast-food restaurant serves an average of 1500 customers per 15-hour day. The manager, using observation at random intervals, counted an average of 75 customers in the restaurant at any time based on occupied tables. How much time does a customer spend, on average, in the restaurant?

So, we have I = 75 customers and λ = 1500 customers per 15-hour day or λ = 100 customers per hour, all on average.

Putting this into Little’s Law, would give us

\[ T = \frac{I}{\lambda} = \frac{75}{100} = 0.75 hrs = 45 mins \]

So, a customer spends on average 45 minutes in the restaurant – including waiting for service, eating, paying, etc.

Notice how it was relatively easy it would be to estimate the average number of people in the restaurant using just two fairly quick to obtain measurements. The result can be used to plan the size of the restaurant, the required number of tables, and staffing. It can also help determine if a marketing campaign is warranted because the restaurant might have empty tables on average. Of course, this is all “on average” and does not describe any particular point in time.

Example III

Brittany is going through Boston Logan Airport security. Her friend told her that it took her 20 minutes to get through security and that 6 lanes are open. How many people would you expect are waiting in line to get through security?

In this example, we have the time it took one person to go through the process, so using that as “an average” (not a super great idea, but that’s all we have), we would have an estimate for T = 20 mins per passenger. Assuming that all lanes are somewhat equal in terms of processing passengers, we could estimate that others took the same amount of time to get through and that the airport would process 6 passengers every 20 minutes. So, we can use that an estimate of λ = 6 passengers per 20 minutes. We could convert the units to hours, but that’s not necessary as they cancel out anyway in the next step where we calculate the “inventory” of the process, i.e., the number of passengers being in some state of processing.

\[ I = \lambda \times T = 6 * 20 = 120 \]

So, we would expect (based on the limited data we observed) that there are about 120 passengers in line or in some state of processing.

Example IV

Patricia Brown, a business analyst at Gray & Partners LP, has conducted an extensive review of their accounts payable process as there’s a process with invoices backing up and not getting paid on time. She discovered that an invoice received from vendors takes, on average, 72 days to be processed and paid. She also found, by looking at records for the past 19 months, that, on average, 529 invoices get paid per month. If an accounts payable specialist can process 3 invoices per hour, how many days of accounting work will it take to go through the backlog of invoices? Assume that a single full-time accountant works about 4.5 productive hours per day. There are 30 calendar days in a month. Further assume that no new invoices arrive and that we are only dealing with the current backlog.

To tackle this problem we need to first know how many invoices are currently in the backlog, i.e., waiting to be paid. We can use Little’s Law for that. In the calculation below, we must make sure all time units are the same – we’ve converted everything to “months”.

\[ I = \lambda \times T = (529 \frac{bills}{month}) * (\frac{72}{30} {months}) = 1270 bills \]

So, now we know that about 1270 bills are waiting (somewhere in the process) to be paid. If 3 invoices are processed per hour then it will take 423.2 hours to process all of the invoices. If an account works 4.5 per day, the we would need 94 days to work through the backlog. So, theoretically, if we hired 94 accountants we would be able to get through it in one day.

The video below takes you through those calculations.

Case Study

A group of IS students conducted an observation of the ordering process for burritos at Qdoba on Huntington Avenue in Boston around Northeastern University. They made the following spreadsheet of their process time-study findings. On a different day they observed the number of customers in line during a 15 minute interval and the average flow rate in that time. For each step in the process they made five observations and then averaged them out to arrive at an average time to make one burrito of 96 seconds.

Using the observations at a different time, we can calculate the flow time using Little’s Law:

\[ T = \frac{I}{\lambda} = \frac{17}{11} = 1.55min = 93sec \]

So, the time study revealed 96 seconds and the Little’s Law calculation resulted in 93 seconds – quite close. This demonstrates that flow time can be calculated either way. Little’s Law is generally less laborious but does not provide any insights into the process. Time studies require more time but collect more granular data.

Lecture

In this lecture, Dr. Schedlbauer attempts to shed some light on the use of Little’s Law for analyzing the aggregate execution time of a complex process.

Slide Deck: Little’s Law for Process Time Analysis

Summary

In summary, Little’s Law is an important alternative to calculating the run-time and other characteristics of a process compared to more labor-intensive time studies. However, Little’s Law provides a macro view of a process and is not useful at determining where a process has inefficiencies and where there are areas for improvement.

All Files for Lesson 42.205

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