Lines, lines, lines!

Regular readers of this blog know that I like eye candy and can rarely pass up a styling graphic. Today’s Wall Street Journal obliges with a great one on a subject near and dear to my heart, queues! (Find the Best Checkout Line, Dec8)

Check it out. It even mentions Little’s Law!

Here is a video of the author discussing his report:

Vodpod videos no longer available.

So I think this is a fun article but there are some points to quibble with.

First, the graphic above describes Little’s Law as a way to determine the average wait but it leaves off an important caveat. Little’s Law is based on long run averages. Thus, it would give the wait you would expect before you get to the store, not the wait you would expect once you see how long the line is. That is, this is helpful in deciding whether you have time to run an errand, not how long it will take you to finish the errand once you are into the process.

Note that Little’s Law uses the arrival rate to the queue, not the rate at which the cashier can process customers.  That’s not a mistake; that is the correct formula. However, for the system to be stable, the cashier has to work faster than work is showing up. If two customers arrive to the line every minute, the cashier has to be able to process, say, three customers per minute. Then if you have six people in front of you, you expect to clear the line in 6/3 = 2 minutes not 6/2 =  3 minutes.

(A more technical aside: Since Little’s  Law gives the overall average line length, it is averaging over even when there are no people in line. Conditional on there being people in line, the average has to be higher. Now suppose the overall average line length is six. If you arrive and see a line of six, you are actually seeing a favorable draw conditional on the fact that you have encountered a line. Hence, you should expect a wait shorter than the overall average.)

A second point. The article makes the (true) point that pooled systems where all cashiers pull from a common line is more efficient. It is referencing the following (very spiffy) video:

Vodpod videos no longer available.

This is all true but I would assert that the extent of the gain (a factor of three!) claimed in the article isn’t going to happen in reality. Why? The analysis behind this calculation is based on an extreme assumption. Specifically, that in the unpooled, multi-line system customers blindly join the line without seeing the overall state of the system, i.e., they don’t consider how many people are in each line. Further, it assumes that people never switch lines when another moves faster.

That’s not how the world works. Not all shoppers are queuing theorists but they know enough to go to an idle cashier if one is available. That will do a lot to close the gap between pooled and unpooled systems.

A final point. Some of the more interesting points in the article refer to psychological research on queuing (as opposed to operations work on queuing). In particular, they cite work from Ziv Carmon. Ziv and I used to toil together in the land of pine trees and ACC basketball. This is not the first time the Journal has reported on Ziv’s research on waiting in line. Check out “Behavior: Merchants Mull the Long and the Short of Lines.” (September 3, 1998)