OK, back to queues and grocery stores. Here is a fun post from a high school math teacher near Santa Cruz. Start with this picture:
Each red brick represents a shopping cart at the grocery store; the numbers are how many items are in the basket. Shoppers are in line. You show up. What line should you get in?The real question here is whether you should patronize the express lane — here the long line with all small jobs. Basic queuing analysis would say that if you are holding just a loaf of bread and a gallon of milk, you should go for the express lane. You can see the amount work in front of you and there are only 11 items in front of you at the express lane but there are 19 in the regular lane. But reality is not so clear. What if there is a set up for each basket? That is, suppose there is a fixed time to checkout that is independent of the number of items (eg cashier greets customer; customer signs credit slip etc). Clearly, if that time is long enough, you may be better behind the big basket with just one set up.
Turns out this is not just a hypothetical question. Dan Meyer, the high school teacher, went a collected a bunch of data and, dare I say, ran a regression. Here is the result:
Sure enough it takes longer to check out a big basket but the set up time is a major factor. If you have fewer than 14 items, the majority of your time in service will be on set up as opposed to having your items rung up!
This is a really sweet example. There are some lucky kids in Santa Cruz to have a guy this clever for a teacher. Here is an interesting follow up. Which customer is more valuable to the store? The shopper with the big basket or the shopper with the small basket? This goes beyond how much they are buying today. I would venture that most shoppers settle on one or two stores for their “main” (say, weekly shop). If you just need milk and bread you may be less picky about where you go. If customers don’t realize there is a set up and those with small baskets naively join the express lane, the big winners here are the big basket shoppers. That is, while it seems like express lanes are a perk for convenience shoppers, they really make regular shoppers better off.