Queues These Days as Reminders of Little’s Law

By Jon Miller Updated on October 19th, 2020

I’ve never been fond of queues. It’s not so much the standing and waiting that bothers me. Instead, it’s the reminder that queue exists because there is an imperfection in the process that prevents smooth flow. Of course, perfection is a dream rather than an expectation. But many process flaws are preventable, easy to fix. They jump out to the kaizen eye.

These days even queues are behaving differently. Many of the physical queues in my life have been shorter, faster-moving, better organized. Stores, restaurants and even public spaces now have occupancy limits. On the other hand, there are notable instances where queues are much longer. It’s interesting to reflect that we are seeing Little’s law in action in daily life.

What is Little’s Law?

Little’s law expresses the interaction between throughput, WIP and lead time in a simple equation. This law is named after the MIT operations research processor John Little who observed the relationship between the number of customers in a store with the rate of their arrival and the average time each customer spends in the store.

The formula for Little’s law is:

Average number of customers = Average arrival rate x Average time in the store

For example, if customers spend an average of a quarter of an hour, or 15 minutes in a store, and 8 customers arrive per hour, the average number of customers in the store at any time would be two people.

8 customers/hour x 0.25 hours/store = 2 customers/store

In commonsense terms Little’s law means as the longer the queue, the longer the wait, unless service is sped up.

Agile Kanban & WIP Limits

In the Agile application of Kanban, Little’s law has been adapted as

WIP = Throughput x Lead Time

One of the key practices of Agile kanban is to set WIP limits for each process stage as well as for the whole system. Once the WIP limit is reached, new tasks can only be started one a task is completed. The completed task acts exiting a process stage acts as a pull signal. Setting WIP limits helps reduce losses from multitasking. It also allows teams to get better predictability of output for inherently variable knowledge work.

Little’s Law Reveals Bottlenecks

Little’s law helps us to become aware of process bottlenecks. In a WIP-limited system, a new task can only enter a process stage after one has left that stage. In other words, the throughput of the whole system is limited by the throughput of the slowest process. Resolving bottleneck processes within kanban or other WIP-limited system is important to prevent idled capacity utilization and slowed delivery performance.

General Process Insights from Little’s Law

In general terms, Little’s law reveals the relationship between its three elements. Changes to one affect the others. For example, when WIP is constant, we must decrease lead time if we want to increase throughput. If lead time through a process is constant, we must increase throughput if we want to reduce WIP. If the throughput rate is constant, we must decrease WIP if we want to reduce lead time.

Understanding Little’s law allows organizations to analyze their process and make changes to methods or policies to achieve our desired result.

It also allows us as consumers to understand the reasons some services are taking longer. Here are a few notable recent examples.

Little’s Law in Mortgage Refinancing Applications

Due to historically low interest rates, mortgage refinancing applications are way up. The banks aren’t adding staff to keep up with this temporary demand. Some banks are setting WIP limits by slowing demand, offering rates that are not as attractive. The WIP, or volume of applications is very high, the processing throughput time is fixed, and therefore the lead times are long.

If this was only a matter of customers waiting longer it would be okay. However, mortgage and refinancing applications have a shelf life. Just like a piece of fresh produce, a seasonal product or concert ticked, mortgage application goes bad if kept in WIP too long. Credit checks or financial documents that applicants provide are no longer usable after they sit in queue at the bank for more than a couple of months. The banks want to minimize their risk that our bank balances, credit scores, etc. may have worsened while the application sat for months in WIP. In my case, the bank assigned sixteen additional tasks for resubmitting or otherwise refreshing our application.

Little’s Law and Building Permit Applications

The building permit application lead time at my local city hall is fourteen to sixteen weeks. That’s the time to wait for them to process it, after collecting all forms and giving it the initial OK. The paperwork takes hours, once they can get to it. Why the long lead time? The built-up backlog of WIP.

Municipalities are receiving high numbers of building permit applications. This is because family members are spending more time working or learning at home. Many of our homes aren’t built to have all family members at home all day. People may need a home office, an addition, or a remodel to accommodate new routines. The inputs are increasing. The city has not set WIP limits for building permit applications.

The city planning department staffing hasn’t kept pace with demand. And in making the shift to working from home, throughput initially declined as they started up new virtual processes. This decline in throughput, rise in WIP and lengthening lead times is a classic application of Little’s law. The city has done an admirable job in making adjustments to the circumstances. Still, four months is a long time to wait for a family who wants to make their home more livable during COVID.

Little’s Law and the Lingering Pandemic

In a way, what we are witnessing with our lingering COVID epidemic in the United States is a failure to understand and apply Little’s law. We are told that the time for the virus to pass through a human is about two weeks from exposure to becoming noncontagious. For a small percentage of people who become ill this is longer. The throughput time is based on the biology of the virus. For all practical purposes, the throughput time is fixed. Humans don’t have control over this, at least at this time.

The lead time is how long this virus sticks around in our society. Lead time is variable and controllable. Because throughput time is fixed, lead time is a factor of WIP. In countries such as New Zealand, South Korea, Vietnam, Japan and China the lead time for COVID seems to have been kept to between 3 and 6 months. In the US the lead time is nine months and counting. The difference lies in how well each set and enforced WIP limits.

The WIP in this example is infected humans. We started 2020 with a very low WIP number. We had the opportunity to set and enforce strict limits, let the virus die out within weeks, and enjoy a return to normal after a short lead time. Unfortunately, we squandered that opportunity by talking a lot about personal liberty and too little about Little’s law.

The ways to WIP limit, or contain the spread of this virus, have become familiar practices: wear a mask, stay at home, get tested, quarantine, and avoid the three Cs of closed places, crowded spaces and close-contact settings. If we want to reduce lead time for a process with a fixed throughput, we must limit WIP. It’s operations research 101.

  1. Akron Rick

    October 20, 2020 - 8:01 am

    I agree with everything but the General Process Insights section. First, none of the three variables are constant; they are all…variable. Second, we don’t even want to hold any of the variables constant; we want throughput to go up, and lead time to go down so that WIP goes down. The equation helps us by pointing out that we shouldn’t focus on any of these variables in isolation of the others.

  2. Jon Miller

    October 20, 2020 - 9:46 am

    Hi Akron Rick

    Thanks for your comment. You’re right, we need to consider all three variables. To clarify, there are times when one of the three factors are constant. Throughput may be limited by your capacity. Sure, you can invest in assets to increase capacity or loss reduction activity. But on a day to day basis, a team’s capacity is fixed. Likewise with WIP. Sure, you can decrease or increase it. But Agile kanban sets WIP limits precisely so that teams can achieve some predictability in terms of lead time. Lead time through a chemical plant, brewery or other process industry is fixed, until equipment is redesigned, formulas are changed, etc. In some processes, all three are changeable at will. In others, one of them is often fixed for all practical short-term purposes.

  3. Benjamin Fishbeyn

    October 21, 2020 - 4:25 pm

    Hello I am a student at the University of Rhode Island. On our campus we have mandatory corona virus testing policy. The queue for these test sites are extremely long and endangered the patients around them. The main issue I noticed is that many of the nurses have to multitask between instructing patients and handing out testing equipment. For this process I would use the Agile Kanban and WIP limits since it helps reduce losses from multitasking.

    • Jon Miller

      October 21, 2020 - 4:44 pm

      Well done Benjamin. I hope they take your suggestion. Stay healthy.

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