This post is part of a series on Queueing Theory.Â There’s also a Little’s Law Reference you can read.

A queueing system is a model with the following structure: customers arrive and join a queue to wait for service given by n servers. after receiving service, the customer exits the system. a fundamental result of queueing theory is little’s law.

theorem: for a queueing system in steady state, the average length of the queue is equivalent to the average arrival rate multiplied by the average waiting time. in other words,

L = Î»W

at amazon, i used little’s law all the time. in dynamic systems with n+ dependencies, it is very helpful to know where the bottlenecks of the system are and how to increase efficiencies, reduce time traps, and eliminate waste in order to increase material flow. in other words, we want product to flow as fast as possible: click-to-ship.

here’s an example:

say there’s a warehouse with 4000 pallets of product that turns ~4 times per year. do we have enough labor to support these transactions? using little’s law, we get

4,000 = Î»(.25year)

so,

Î» = 16,000 pallets/year

assuming a 10 hour shift per day of about 250 working days per year, there is roughly 2,000 working hours. this means, then,

Î» = 8 pallets/hour

the analysis above is critically important to estimate the labor force required to move pallets, receive product, move product, and get work done, in general.

there are many more applications of queueing theory that i will explicate and share in the next while. queueing theory is a critical, underused, but very valuable principle in business.

www.comp-nerds.co.nr says

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Mark Graban says

Little’s Law is an incredbly helpful tool for operations. I highly recommend the Hopp and Spearman book “Factory Physics”, which spends a lot of time on Little’s Law.

Another way of thinking about this is:

Throughput = WIP / CT

WIP = Work in Process

You have to be careful with Little’s Law and resist the temptation to think “all I have to do to increase Throughput is to increase WIP”. It doesn’t work that way, not that Peter was implying that. I’m just saying that’s a common mis-application of Little’s Law.

When you start with zero WIP, you will have zero throughput. Makes sense, right? At first, as WIP increases, your throughput will increase somewhat linearly. But, as capacity utilization reaches 80% or so, throughput levels out and Cycle Time explodes. That’s why really busy systems (factories, emergency rooms, etc.) have very long waiting times. The Cycle Time explosion is made worse with variation. The more variation you have in customer demand and service time, the worse the Cycle Time explosion is. That’s why leveling (heijunka) is key…. on to Peter’s second post, I’m looking forward to reading it.

This Industrial Engineer is just happy to see anyone discussing Little’s Law.

Mark Graban says

One problem with your analysis there is that you’re assuming demand (and labor needs) are constant throughout the year. I’m certain that wouldn’t be the case for Amazon or any retailer.

Richard Karpinski says

“assuming a 10 hour shift per day of about 250 working days per year, there is roughly 2,000 working hours.”

This works only if the 10 is in octal notation.

LT says

10-hour shift, but 8-hour working. Hey, this is normal.

Bob Dierks says

Refreshing!

Baruch Atta says

“…as capacity utilization reaches 80% or so, throughput levels out and Cycle Time explodes. Thatâ€™s why really busy systems (factories, emergency rooms, etc.) have very long waiting times…”

No, that’s not the reason. The reason that ERs, restaraunts, and Post Offices have long wait times is that utilization (or demand) has gone over 100% throughput capacity. The service can no longer serve in the rate as people arrive. Since the service provider has not planned for or reacted to the higher demand, the length of the line grows.

Baruch Atta says

should read…

…can no longer serve at the rate that people arrive…

Israel Robert A. says

“….The reason that ERs, restaraunts, and Post Offices have long wait times is that utilization (or demand) has gone over 100% throughput capacity. The service can no longer serve in the rate as people arrive. Since the service provider has not planned for or reacted to the higher demand, the length of the line grows”…….

it’s not always the case that when demand or utilization goes over 100%, long waiting lines develop in busy systems.

this could also happen when there are system inefficiencies so that average service time< average arrival time.