Queueing
Little's Law does not care what the board says
One claims desk, two honest answers to how long a claim takes. The difference between them is the queue, and the arithmetic that finds it fits on a single line.
Ask an adjuster how long a claim takes and you will be told how long it takes to work one. Ask the customer waiting on it and you will be told something else entirely. Both of them are answering honestly and neither of them is describing the same quantity. Only one of them is describing the queue, and the queue is where the days go.
The formula that connects the two is the shortest useful thing in operations. John Little published the proof in 1961, in a paper titled “A Proof for the Queuing Formula: L = λW”. Three symbols. The work in progress in a system equals the rate work arrives multiplied by the time each item spends inside.
What makes it worth knowing is not the algebra. It is that the law holds without asking anything about the process. It does not care how many steps there are, in what order they run, how the work is prioritized, whether the queue is first in first out, or what the work actually is. Rearranged, it says something a manager can act on this afternoon: the average time an item spends in your process is set by how much work is sitting in it and how fast work is arriving. Nothing else.
The desk
Take a claims desk. Over four weeks, twenty working days, it received 680 new claims. Somebody counted the open claims at close of business every one of those days and the count averaged 122.
W = L ÷ λ
| L | Claims open on the desk, averaged over the twenty days | 122 |
|---|---|---|
| λ | Claims received, 680 over twenty working days | 34 a day |
| W | Average time a claim spends on the desk | 3.6 days |
The figures here are a worked example rather than a measurement taken from a real desk. Both inputs are things you can count on your own with no instrumentation at all: a daily tally of what is open, and the arrival log you already have.
Three and a half working days. Now ask the desk what a claim takes and you will hear forty two minutes, because forty two minutes is how long an adjuster spends on one. That number is also correct. It is the working time, and it is what gets reported upward, because it is the number the desk controls and the one it is proud of.
The customer never experiences forty two minutes. The customer experiences three and a half days, and the difference is not effort, skill or headcount. It is queue.
The lever nobody reaches for
Read the formula again with an eye on what you can actually change. Arrivals are the customer’s decision, not yours. The time in the system is the thing you are being asked to improve. That leaves work in progress, and work in progress is the only term on the right hand side that is entirely inside your gift.
So cap it. Suppose the desk decides that no more than sixty claims may be open at once, and anything beyond that waits in a holding queue before it is opened and assigned.
W = L ÷ λ
| L | Claims allowed to be open at once, by policy | 60 |
|---|---|---|
| λ | Claims received, unchanged | 34 a day |
| W | Average time an opened claim spends on the desk | 1.8 days |
Same staff, same skills, same arrival rate, same working time per claim. The only change is how much work is allowed to be in progress at once.
Nothing about the work changed. No one worked faster. The desk did not get bigger. What changed is the amount of work sitting in the middle of the process at any moment, and the time in the system halved because the arithmetic gives it no choice.
There is a real cost, and pretending otherwise is how this idea gets oversold: the claims held back are still waiting, they are just waiting in a different place. What the cap buys is that the work which has started now finishes quickly and predictably, which is worth a great deal to the person who has to answer the phone, and it makes the true demand visible instead of hiding it inside a pile of half worked claims.
Two ways to get this wrong
Mixing the units. The law is unit blind, which means it will hand you a wrong answer with total confidence if the arrival rate and the time are measured on different clocks. Count the same desk’s arrivals per calendar day instead of per working day and you get 680 over twenty eight days, which is 24.3 a day, and the answer becomes 122 divided by 24.3, or 5.0 calendar days. That is the same answer, not a different one: three and a half working days across a five day week is five calendar days. If the two figures had not reconciled, one of them would have been measuring something the other was not.
Measuring during a pile up. Little’s Law describes a system over a window where what goes in roughly matches what comes out. Measure it while the backlog is building and the answer flatters you, because some of the work counted in the arrivals has not left yet and its long wait is not in the average. The check takes a minute: compare total arrivals with total completions over the same window. This desk received 680 and closed 668, a difference under two percent, so the window is stable enough for the answer to mean something. If it had received 680 and closed 500, no arithmetic on this page would have been worth reading, and the honest report would have been that the desk is losing ground at 180 claims a month.
What it is for
Little’s Law does not tell you how to improve anything. It tells you what has to be true, and that is more useful than it sounds, because most improvement arguments are made without checking whether they are arithmetically possible.
Somebody proposes a project to cut handling time by fifteen percent. Handling time is forty two minutes, so the project is arguing about six minutes inside a three and a half day journey. Somebody else proposes a rule about how much work may be open at once, which sounds like bureaucracy and costs nothing. One of those two changes moves the number the customer feels. The formula tells you which, before either of them is funded.
It also sets the ceiling on any promise you make. If you want to quote a two day turnaround at thirty four claims a day, you are committing to holding open work at about sixty eight claims, permanently. That is not a target to be exhorted toward. It is a constraint, it follows from two numbers you already have, and if the desk cannot hold it then the promise cannot be kept, whatever anyone puts in the plan.
The next question, once the queue is visible, is where inside the process the waiting actually collects. That is a different measurement, and it is the one that separates cycle time from lead time.