In physics, laws are a fundamental part of understanding how systems operate. E=mc2 would have to classify as the most famous example. From these three simple letters, physicists and others have been able to greatly improve their understanding of how the universe works and make predictions about how things will behave under differing circumstances – pretty powerful stuff.
Such laws exist in the field of operations management as well. One in particular deserves to become much more familiar to managers. It is called Little’s Law, named after the man who first mathematically proved it, John D.C. Little, former Institute Professor and Chair Management Science for MIT’s Sloan School of Management1.
While simple on the face of it, Little’s Law may be thought of as ‘the master of cycle time’ because it defines the critical relationships that dictate how long it will take, on average, to complete the work tasks in a particular process. Any process. For example, how long it takes to complete a manufactured good, process a customer order, serve a patient in an emergency room or complete a construction project? And time, especially cycle time, is money. By establishing the critical relations driving cycle time, Dr. John Little provided the key to understanding process efficiency.
A Look at the Law
Littles Law states that “The average number of customers in a system (over some interval) is equal to their average arrival rate, multiplied by their average time in the system.”2 Symbolically this can be represented for our purposes as:
WIP = TH * CT
- TH = Throughput (arrival rate). This is the velocity or speed of production. It is calculated by determining how many items are produced and dividing this by the length of time it took to produce them, but it can of course, be computed from Little’s Law; TH = WIP/CT
- CT = Cycle Time (average time in the system). This is the time it takes to complete the production cycle or the average time it takes to produce one unit. Generally, cycle time requires either direct measurement to determine (technically very difficult) or can be computed from Little’s Law (one of the powerful things about this Law as we shall see); CT = WIP/TH.
- WIP = Work in Process (average number of units/customers in a system). This is the number of items currently in production or being serviced in some way. Again, this figure must be measured (counted) directly or can be computed from Little’s Law.
Does Little’s Law look to basic, too small to have any practical significance? Well, Littles Law is now a fundamental part of queuing theory and has found broad application in the design of computing systems, customer service functions and logistics. But it has much broader applications. In fact, even if you are unaware of Little’s Law, it may already be changing the way you think about operations and production. This, because organizations pursuing Lean production methods, are pursuing an improvement strategy that is based on the reality of Little’s Law. Lean assumes Little’s Law works.
A Production Example
So what does Little’s Law say and what are the implications for managers? Well, first let’s consider a typical production situation – accepting new orders into a production process. Let’s assume we are running a process where throughput (TH), the number of units we produce, is equal to 25 units per day. Our work in process (WIP), the number of units in various stages of production, remains relatively constant at 100 units. Given these conditions, our cycle time, the average time it takes to complete one unit, would be 4 days (CT = WIP/TH, CT = 100/25, CT = 4).
This means we can accommodate new orders of 25 units each day and the system remains in balance. But suppose one day, we receive orders for 60 units – 35 more than the standard order of 25. WIP would increase from 100 units to 135 and with TH remaining constant at 25 units per day, CT would immediately increase from 4 days per unit to 5.4 days per unit. The increase in orders (normally considered a good thing), immediately causes a decrease in production efficiency (a bad thing). This is one of the strange but accurate implications of Little’s Law, significant levels of new orders cause production efficiency to decrease.
Adding to the confusion is that delivery promises made to customers at the time the orders are taken are typically based on historical cycle times – in this case 4 days. But the very act of taking these orders has increased the cycle time by 35%, making it impossible to meet the delivery times promised.
Beyond the obvious results of missed delivery dates and cancelled orders, the impact of this may include an increase in inter-departmental squabbling. Marketing blames ‘slow’ Operations for the missed deliveries and Operations blames Marketing for ‘over-promising’ on delivery dates to make sales. Both are victims of Little’s Law. If your organization is confronted with sources of friction such as this, you may want to conduct a little operations research to evaluate the extent to which Little’s Law is at work in your organization. Chances are you’ll find it is — with a vengeance.
Implications for Project Management
Little’s Law applies the same way to problems in project management as it does to production. Instead of producing units, we are now concerned with completing projects.
For example, an Information Technology (IT) department may take on additional projects for client departments (increasing WIP) without realizing this immediately causes the project completion time (CT) for all projects to increase. This may explain why so many IT projects take longer than expected. New projects added to the department’s list of active projects immediately causes the time it takes to complete each individual project to increase.
The same holds true for construction firms taking on new or significantly larger projects or companies increasing the number or scope of improvement/change initiatives such as those employed by Six Sigma. In these cases, increasing the number of projects (WIP) would immediately yield a decline in performance in the form of increased cycle time with every project taking longer to complete.
The result is a general slow down in the delivery of all projects as resources become too thinly spread and production or operational bottlenecks emerge. The result is often failure to meet delivery times (even with projects already underway and not associated with the ‘new orders’), declining operational performance, overall project failures and cancellations.
Little’s Law Everywhere
Little’s Law applies to any system, not just production, manufacturing or project management applications. Once you become familiar with it, you begin to look at everything just a little differently.
For example, a social services agency may run a successful counseling program that lasts 5 weeks (CT). At any time there are approximately 50 people enrolled (WIP) in the program which means the program, is graduating about 10 people per week (TH). This then, is the maximum number of new clients the agency can accept each week if it is to keep its program intact. If, because of its success, it takes on more clients in a specific week than is typical, lets say 15, the cycle time will increase to 5.5 weeks. Schedules will immediately become problematic and the quality of the program will decline for all those enrolled. This explains at least in part, how it is so many successful programs become the victim of their own success (why good programs go bad).
Here is a favorite of mine. Consider the new wave of enterprise-wide accounting and information processing systems that organizations are purchasing today, often at the cost of millions of dollars. The general management consensus is that these new systems will make for faster and improved decision making by making more information, of better quality, available to decision makers. Well maybe. However, an argument can be made that such systems also increase the amount of information (WIP) to the decision making process. With this increase in WIP, the time it takes to make decisions (CT) will increase. Simply stated, by making more information available for processing, these new systems will ensure that decision making actually takes longer and that arguably, the quality of decisions made, will decline.3
Other examples abound. What are the implications of increasing classroom size in public schools, the impact of growth on corporate resource departments such as human resources and finance, or the impact of adding additional product lines or pricing schemes on order processing? Plugging the numbers into Little’s Law often yields interesting and surprising results.
The Law is the Law
Most organizations and managers are unaware of Little’s Law. Those that are aware of it, tend not to believe it. The implications are at times, just too counterintuitive. But the law is the law and Little’s Law is one you ignore at significant potential cost.
An increase in work, be it in the form of new or bigger projects, or orders, or number of clients or customers, will tend to increase cycle time for all items currently in the system causing a decline in system performance across the board (assuming throughput stays constant). You cannot escape it.
To improve cycle time, only two options are available, increase throughput or reduce work in process. The first option, increasing throughput, may require process improvement or significant investment to increase the scale of the system to better handle the increase in work in process. This is fine when considering longer term system capacity. It is impractical, however, when addressing relatively short term variations in demand.
Only the second option, reducing work in process, can be used to effectively address these short-term demand variations. Reducing work in process may require some counterintuitive actions. Among these are temporarily pulling projects or orders out of the work flow and setting them aside. With the resulting decline in WIP, cycle time (CT) drops and the remaining projects get done better and faster. So much so, that projects originally pulled out of the workflow, can then be reinserted, often being completed on or before the original planned target date. In other words, by stopping work on a project, it gets done faster. Now that is counterintuitive! 4
Don’t believe it? Consider this example from Boeing concerning the production of the C17 Globemaster.5 In the early years, production of the C-17 was fraught with problems. There were quality issues, significant cost overruns and aircraft were constantly delivered late to the customer, in this case, the U.S. Air Force.
Exhibit 1 details delivery times relative to schedule for every C-17 delivered to the Air Force for the years 1992 to 2000. Negative numbers indicate delivery was behind schedule, positive numbers indicate delivery was ahead of schedule. Notice the negative numbers in 1992 and 1993, indicating late deliveries (slow cycle times). The improvements in 1993 and early 1994 gave program managers at Boeing some hope – they felt they were making progress and deliveries would soon be on schedule.
But when it became apparent that plane number 12 was going to be significantly behind schedule, Don Kozlowski, General Manager of the C-17 program, came to the realization that to have any hope of meeting the delivery schedule to the Air Force, things were going to have to be done differently. What he did was revolutionary; temporarily take airplanes out of the workflow (reducing WIP) so that cycle times improved. In other words, Don reduced the time it took to assemble a C-17 by stopping work on selected C-17’s in production.
At the time, aircraft moved through various stations on the assembly line with a specific set of tasks conducted at each station. Keeping the aircraft moving through the stations was paramount. Even if all the tasks at one station were not completed, aircraft were moved to the next station where personnel played catch-up in assembly. Boeing believed it had to keep the planes moving to meet the schedule.
What Don Kozlawski did was make quality king, dethroning the schedule as the critical driver of production. He decided no plane would move forward in the production cycle until all tasks associated with that station were completed and completed well. This meant there would be stations where assembly teams would have nothing to do while they waited for the station ahead of them to complete their tasks. These planes ‘in waiting’ were essentially taken out of the workflow (temporarily reducing WIP) while work progressed on the plane holding up production. Once the plane holding up production was capable of moving ahead to the next station, all planes in the workflow would move forward and work on them would proceed.
Exhibit 1 Delivery Time Relative to Schedule
The success of this approach is evident in Exhibit 1. While plane number 12 was late, the plane behind it, number 13, was delivered to the Air Force just slightly ahead of schedule and the program has never looked back since. Cycle time to budget subsequently skyrocketed (as did quality) with C-17’s being delivered to the Air Force in advance of schedule by as much as 170 days. By stopping work on aircraft in production, Don Kozlawski and his eventual replacement, David Spong, assured they were completed not only better, but faster.
Neither Don nor David were thinking of Little’s Law when they made the decisions they did. But aware or not, both were taking advantage of its fundamental principles, reducing work-in-process improves cycle time – period.
Little’s Law and Lean
Many organizations today are pursuing improvement strategies based upon Lean Methods or Lean Thinking. Few realize, however, that Lean Methods are constructed on the foundation provided by Little’s Law. That may be why so much of Lean Thinking, like Little’s Law, is counterintuitive.
Traditional management thinking argues for large production volumes to gain economies of scale. To do so, companies produce large volumes of parts that are later assembled, also in large volumes, into finished products. The economics of this are viewed as obvious – make lots of a specific item, and you can make it cheaply.
Not so obvious to those familiar with Little’s Law, however. Lean thinking argues that these traditional management practices also yield large levels of work-in-process. With that increasing work-in-process comes increasing cycle times and falling levels of system performance/efficiency. This means that large production volumes only look economic when measuring the relatively narrow costs of production for each individual part. When looking at the larger system, including the costs associated with holding inventories, the total cost of production tends to rise with these large production volumes.
That is why Lean thinking pursues methods such as single piece flow, pull systems, Kanban, takt-time production targets and complexity reduction – these approaches are all designed to produce low levels of work in process and in keeping with Little’s Law, improved cycle times.
Taking Advantage of Little’s Law
So how can management take advantage of Little’s Law? How can you use it to gain productive or competitive advantage?
The first step in taking advantage of Little’s law is to accept it. As mentioned, managers that are aware of Little’s Law tend not to believe it. Like the law of gravity, however, Little’s Law is not influenced by whether you believe in it. You can refuse to accept it, and watch your production (or project completion) efficiency fall like a stone. Or, you can accept it, using it to balance system capacity with new arrivals and thereby maximize the short term efficiency of the system.
Determining Cycle Time
A second way to take advantage of Little’s Law is simply in the calculation of cycle time. Determining cycle time is an important and often critical factor in many process performance improvement initiatives. Unfortunately, it is difficult to measure. For example, to determine the cycle time of teller service at a bank, the time of arrival of selected customers would have to be captured as well as the time of the transaction completion. This is an almost impossible thing to do, unless of course, the bank would be willing pay someone to stand at the door with a stop-watch and time the complete customer cycle from arrival to exit. Some customers might feel a little put-off by this. Many other processes or tasks within processes present similar issues in measuring cycle time – it is difficult to identify and capture the time associated with a unit’s arrival or exit from the system.
With Little’s Law, however, you don’t need to measure cycle time directly. If you know how many units (orders, people, and projects) are in the system (WIP), and how many of these are completed during some specified time period (TH), cycle time can easily be computed. Generally speaking, WIP and TH tend to be readily available, one being a simple count of how many units are in the system and the other a count of how many are completed during a day or week or whatever time frame is of interest. In short, Little’s Law provides an easy and accurate way of answering the question: “How long does it take?” (See Box: How Long Does It Take?)
Seeking Further Opportunities for Improvement
Third, some comfort can be taken by the fact that if you or your organization is pursuing Lean Methods, you are already taking advantage of Little’s Law, even though you may not be aware of the Law itself. Knowing what is behind many of those Lean Methods, however, makes it easier to see new opportunities to apply the concepts of Lean to new and different situations
A good place to start looking for those new and different applications is wherever there is some form of production system that must respond to changes in input volume (expressed as arrival rates, new order rates, new projects, etc.).
Here is a example with which I have some personal experience. Some years ago, I was contracted to assist some process improvement teams in their endeavors at enhancing the efficiency of a large scale natural gas processing plant in southern Alberta. Part of this exercise included helping teams properly analyze and interpret plant data in the form of control charts (Statistical Process Control).
A major issue for this plant, as with any gas plant, concerned the number of shut-in’s. A shut-in occurs when the gas being produced is of insufficient quality that it is denied access to the outgoing pipeline distribution network – thus being ‘shut-in’ the plant. Without the usual outlet, and with the gas having to go somewhere, the plant is forced to flare the gas produced by the plant — which is a nice way of saying they burn it. Thus, hundred of thousands of dollars worth of production literally goes up in smoke.
In reviewing a few basic control charts, one process improvement team noticed that shut-in’s (or near shut-in’s as measured by gas quality) occurred shortly after any sudden increases in the volume of raw unprocessed gas flowing into the plant. These increases in input volumes were driven by customer demand. When sufficient new orders for gas came in, field staff were directed to ‘open the taps’ of field wells that led to the plant. The resulting increase in volume would then drive the plant to a shut-in position even when the change was well within the plants operating capacity. It wasn’t the volume of input gas that was creating the problem; it was the variation in input.
We quickly recognized this as just another dimension to Little’s Law. Even though some of the assumptions of Little’s Law were arguably violated in this case, the system represented by the gas plant was still operating in accordance with the Law’s fundamentals. Specifically, a change in demand (variation) was slowing the processing cycle driving efficiency down to a point where the plant was essentially ‘crashing’. The solution was simple, limit the amount of new input gas that could be turned on within a 24 hour period. A simple protocol reflecting the new rules was developed that day and communicated to field staff. Plant shut-in’s fell from several a year to zero.
So what systems might benefit from consideration of Little’s Law? Well here are a few ideas to get those creative juices flowing.
- What is the impact on hospital or medical system efficiency given changes in arrival or referral rates? To what degree are cycle times within a hospital or emergency department impacted by differences in arrival rates? What are the implications for medical service design? In the context of Little’s Law, many of the initiatives designed to improve hospital efficiency seem to be destined to do just the opposite.
- How are engineering and construction (E&C) companies impacted with relatively sudden increase in the number or scope of construction projects? For example, Little’s Law infers that estimation processes based on individual project costs would not be able to assess the impact of having multiple (or large scale) projects as work-in-process. Thus we should see the ability to effectively estimate required project resources decline with the increasing scale or number of projects in-process. I’ll bet this has been the (unspoken) experience with many E&C companies.
- What is the impact of sales ordering processes on the ability to deliver the products or services sold? We have already discussed that sudden short term increases in sales volumes will decrease efficiency and that this decrease in efficiency will virtually guarantee that delivery promises made to the customer cannot be kept. The result, internal bickering among departments and lost or dissatisfied customers. Do your scheduling or ordering systems specifically take Little’s Law into consideration? If not, you are likely sacrificing production efficiency and customer satisfaction. How about that new marketing initiative? If successful, what will be the impact on the time it takes to deliver the product? If you are not careful, that new initiative may actually destroy more customer relationships than it creates.
A Concluding Comment
Little’s Law was originally formulated to address system performance the in the context of customer service – specifically the ability of a system to serve a customer given changes in arrival rates. Subsequently it has found broader application to any ‘production’ system that is dependant on input volume. These are as diverse as patient scheduling, gas plant operations, project management, call center management and manufacturing and production planning.
All of which means Little’s Law deserves still wider recognition and application. Managers, and those concerned with efficient operations generally, are well advised to heed Little’s Law inescapable constraints – cycle time is dependant upon work in process and throughput.
How Long Does It Take?
It is surprising how many people, including managers, executives and others involved in performance or process improvement initiatives confuse throughput with cycle time. How many times have we heard some form of the question: “How long does it take?” For example, how long does it take us to process a customer’s order, to serve a customer, to move components from inventory to a manufacturing station, to process a payment or to complete a task? These are common and essential questions, important for any process owner to know. They all demand a response in terms of cycle time.
However, the answer provided is often a variation of throughput. This occurs when the answer is calculated by taking throughput (say 100 units a day) and dividing by the available time (8 hours in a day) to provide an answer 12.5 units per hour or 1 unit every 21 minutes (100 units divided by 8 hours). But this isn’t cycle time, it is takt time or the inverse of throughput. In providing this answer, we are giving the impression that it takes 21 minutes to produce an item from start to finish but this isn’t true at all. In fact, given the data presented, we have no idea what the cycle time is.
To find the cycle time we would have to physically time units from the point they entered the production cycle to the time they left the production cycle. Or, of course, we could use Little’s Law. To do so, we would also need to know the level of work in process. If this was say, 800 units, the cycle time would be 8 days (CT = WIP/TH or CT = 800 units/100 units per day or CT = 8 days). This is a far cry from 21 minutes.
1. John D.C. Little “A Proof for the Queuing Formula: L = λ W,” Operations Research, Vol. 9, No. 3, 1961 pp. 383 – 387
2. John D.C. Little “A Proof for the Queuing Formula: L = λ W,” Operations Research, Vol. 9, No. 3, 1961 pp. 383 – 387
3. I am willing to wager that this point will not be readily accepted by those selling such enterprise-wide software systems or those than have recently spent considerable sums for their acquisition. It is fair to say that the full process of management decision making is considerably more complex than I have painted it here and in that context, enterprise-wide systems may yield considerable advantages over older systems. My point, however, is that such systems present the very real opportunity of yielding precisely the opposite results of what they are designed, in part, to do and that these possibilities are rarely identified. Little’s Law can help us identify where we should begin looking for problems.
4. Those familiar with the work Dr. Edwards Deming should see some immediate connections between the impact of Little’s Law and Dr. Deming’s observation that increasing variation yields a decline in system performance. In this case, variation in the form of a short term increase in demand yields an increase in cycle time. If this variation were to take the form of a decline in demand, cycle time would decrease but costs per unit would increase as system capacity would be under utilized. The most efficient system in the short term, therefore, is that which manages to minimize variation in demand or arrival rate, keeping it is close as possible to the operating capacity of the system. This is also the foundation of the Lean Thinking concept of takt-time and adjusting the arrival rate of new orders such that the production system remains balanced or synchronized with demand. Takt time is in fact the inverse of the arrival rate or throughput (TH) in Little’s Law. See Lean Thinking by James P. Womack and Daniel T. Jones, Free Press, New York, 2003 for a good set of discussions on takt time.
5. I am grateful to Mr. David Spong, President of Boeing Military Aerospace Support for providing permission to use this example from the C-17 program.
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