Is the express lane in the grocery store always the fastest lane?
That’s a great question and its the subject of a “Dear Mona” column over on FiveThirtyEight (Dear Mona, Which Is The Fastest Check-Out Lane At The Grocery Store?, Oct 16). Mona attacks this question by heading into the queuing theory weeds.
I couldn’t find much research on express lanes specifically, but one paper from Amsterdam found the reduction in wait times for express-lane customers didn’t offset the overall increase in wait times for everyone. Maybe life would be easier if the supermarket didn’t have an express lane — or, better yet, if it got rid of multiple lines altogether and had all customers join a single infinitely sprawling line where there were no winners and losers. That might sound nightmarish, but the math actually suggests it would be anything but.
That math comes from queuing theory, a subject of study that’s been around ever since Danish mathematician Agner Krarup Erlang discovered a method for managing telephone traffic in 1909. To answer your question, I’ve had to take a crash course in (more modern) queuing theory, including examining formulae that calculate how average wait times at the grocery store vary depending on the type of line you join.
I should state upfront Mona on the whole acquits herself quite well on this. But there are a couple of points worth mentioning. First, there in fact supermarkets that run with a single queue, like this Hannaford’s in West Lebanon, NH.
As you can see, that singe serpentine queue ends up chewing up a lot of space at the front of the store. That’s a lot of real estate to give up when you only have two people in line. As we have written about before, that is only one of the complications of having a single queue in a grocery setting.
But let’s suppose for the moment that we can get a single queue to work. Is that in fact the best way to run a supermarket’s checkout?
To answer that question, we first have to specify a decision criterion. Let’s go with minimizing the wait for the average customer. That is a pretty standard approach in queuing theory and is more or less what Mona is using in her article. (More on that in a bit.)
If that’s the goal, then I would indeed advocate a single line. However, that doesn’t go far enough. We have information on shoppers that we can use to reduce the average waiting time. Specifically, we can see their basket size and use a service discipline that accounts for how long it will take to serve them. Service discipline refers to how we take people out of line. Supermarkets default to first in, first out (FIFO). FIFO has obvious, appealing properties. It is both intuitive and fair. But it leads to longer lines.
To see this last point, suppose the store is about to close and we have a single cashier open. There are three final customers that need to be rung up and they have different sized baskets. Mister Big has lots of items and it will take 6 minutes to ring up. Mr. Small has few items and can be rung up in 2 minutes. Ms. InBetween has a medium-sized basket and will need 4 minutes. The first thing to note is that if we look at how long the cashier will be busy, the order we process them doesn’t matter. She needs 12 minutes to get everyone out of the store regardless of how they are ordered.
If instead we look at how long customers are waiting, the order matters a lot. Suppose they are ordered as Mr. Big, Ms. InBetween, and then Mr. Small. Mr Big enters service and doesn’t wait. Ms InBetween waits 6 minutes while Mr. Small waits 10. The average wait across the three of them is then (0+6+10)/3 = 16/3. However, if we reverse the order, Mr. Small doesn’t wait, Ms. InBetween cools her heels for 2 minutes, and Mr. Big is in for 6 minutes. That gives an average of 8/3. Reversing the order cut the average wait in half! If we use FIFO, we will get an average wait somewhere between these extremes. It won’t be as bad as always putting the biggest baskets first, but we will never trump letting the small baskets jump the queue.
So could a supermarket use a basket-based priority? It would be hard. I’ll go out on a limb and say that you risk alienating those who shop for a large family once a week. You lose the perceived fairness of FIFO, and with it the business of every family of four or more. But suppose that were not an issue, and that everyone was willing to play by my rules. It still going to be hard to pull this off. The single queue pictured above is going to need even more space so people can maneuver to the front of the line when it’s their turn.
The current system of express lanes is then a compromise. It would be optimal to pool all the open registers and priorities small baskets (actually, sort everyone by their basket size). Given that we can’t do that, we set aside some capacity for just small baskets.
But what if the goal isn’t to minimize the average waiting time? This is actually the biggest flaw in the article. The question that is asked isn’t really about the best way to organize queues. It’s much more about what line should someone with a small basket get in. Is it always right to join the express lane? When you reach the checkout, the average time the system delivers doesn’t matter. What matters is the time you expect when choosing from the particular lines in front of you.
Fortunately, we’ve covered that before. The key point is that a basket of 20 items does not, in general, take twice as long as a basket of 10 items. Payment, whether swiping a credit card or counting out change, doesn’t really vary with basket size. We can consequently approximate the processing time of a customer by linear function with a positive intercept.
What that means is that a customer getting to the checkout lanes should pick the line with the lowest workload to process. The workload would account for each item that needs to be scanned and bagged as well as the number of payments that need to be taken. Said another way, you’re better off being behind one person with a 40 item basket than two people with 20 items each and being behind those two is better being behind eight people with 5 items each. Thus foregoing a longish express line for a shorter regular line may make sense even if those in the regular line have pretty full carts.
Finally, thanks to Kerry Driscoll from my MA 324 section for sending me this article.
