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Little’s Law and Queuing Theory

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Anyone who wants to improve productivity needs to know a little queueing theory. That mouthful to say is luckily not too hard to understand.

Queueing theory is the analysis of queues. Or, in simpler terms, it is the study of lines.

Why would anybody want to spend hours thinking about waiting in lines? In order to find ways to make those lines go faster, of course.

And if you think about it, most of the routine work you do every day in the office, the shop or even in the garden involves stuff waiting in lines. Usually, though, the most important queue to worry about is a queue full of people. Cartons of milk might expire but individual people hate to wait in lines.

The question which comes up all the time in queuing theory is the one you ask when you arrive at the restaurant: how long is the wait? That’s the most important piece of information to you, the potential customer who is ready to eat. However, a productivity expert has to start with a slightly different question: how big is the line?

Consulting on Queueing

© Flickr user hktang

Now here’s what’s absolutely amazing about queuing theory. It might seem like this is insanely complicated question. Doesn’t it depend on the time of day? Doesn’t it matter how long people sit at the restaurant? Isn’t it affected if the kitchen is understaffed?

Nope. As long as the system is a steady state (it’s not ramping up or shutting down), it’s easy to understand the total size of the line. It’s an amazingly easy formula:

 size of the line  =  rate of arrival  ×  average time in the line 

This is called Little’s Law, and it’s the foundation of queuing theory. So how about an example with everyone’s favorite topic, email?

Suppose you are getting ready to go on vacation next week. This is a good time to figure out how much email you are getting and how much you are processing, so you decide to see how much email you have in your inbox at the end of every day.

Each day, you make a note of the number. And then you add them all up and take the average at the end of the week. Let’s say your typical inbox size after adding them all up and diving by five days in the week is 247 messages.

You head out of town for the weekend and don’t return until Wednesday morning. Now, there are an additional 120 messages that you missed while you were out. Since you were gone two days (or four if you count the weekend), that’s a rate of arrival of 60 messages per day. Now you know the details:

(Average) Size of the line: 247 messages
(Average) Rate of Arrival: 60 messages per day
(Average) Time in your Inbox:  ?

Did you do the arithmetic yet? That’s right, the average time a message spends in your inbox is 4.11 work days!

(Of course, you don’t have to go on vacation to get this data. It’s just easier to count rate of arrival if you can ignore the inbox for a while.)

If you can convince your coworkers to send less mail or reduce your average inbox size, you can make a big impact on your personal productivity. If you can get your inbox down to ten messages, you’ll be able to respond to every email the same day! (Yes, we’ve talked before about getting down to inbox zero with email management.)

So that’s an introduction to queuing theory. You can apply to it any line in your office: whether it’s a sequence of handoffs, processing paper work or even the flow of telephone calls. Try it yourself or contact the Indianapolis productivity consultants at AccelaWork. We’d love to hear from you!

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Robby Slaughter
Robby Slaughter is a workflow and productivity expert. He is a nationally known speaker on topics related to personal productivity, corporate efficiency and employee engagement. Robby is the founder of AccelaWork, a company which provides speakers and consultants to a wide variety of organizations, including Fortune 500 companies, regional non-profits, small businesses and individual entrepreneurs. Robby has written numerous articles for national magazines and has over one hundred published pieces. He is also the author of several books, including Failure: The Secret to Success. He has also been interviewed by international news outlets including the Wall Street Journal. Robby’s newest book is The Battle For Your Email Inbox.
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