You’re at the supermarket. Which line should you choose for fastest service?
(The numbers are the number of items in each cart.) Dan Meyer wondered about this, and rather than merely speculating or diving into queue theory, went out and collected data. (A very Mythbusters attitude, of which I approve.) He spent ninety minutes watching the checkout lines at his local supermarket, counting the number of items in each shopper’s cart and the amount of time it took them to be checked out (from loading up their first item to completing the financial transaction), and the method of payment.
The conclusion? In the example above, you’re likely better off in the shorter line with one loaded cart, rather than the “express” lane with several carts with a few items. The reason is that it takes about 3 seconds to scan each item, but on average about 35 seconds to process each shopper. In the example above we have 11 items and 4 shoppers in the express line for a wait time of 176 seconds, versus 1 shopper and 19 items for a wait time of 96 seconds. Here’s the calculation, as done in R:
> shopping <- read.csv(
+ “http://spreadsheets.google.com/pub?key=tE9pXlYLwTAeiDWxL8h_viA&single=true&gid=0&range=A1%3AE37&output=csv”,
+ as.is=TRUE)
> shopping$seconds <- as.numeric(as.difftime(shopping$Total.Time))
> p1 <- coef(lm(seconds ~ Number.of.Items, shopping,subset=-8))
> p1
(Intercept) Number.of.Items
35.309942 3.191313
> p1 %*% c(4,11) # express lane
[,1]
[1,] 176.3442
> p1 %*% c(1,19) # short lane
[,1]
[1,] 95.9449
Note that there’s one outlier (row 8) in the data file which I’ve deleted (and Dan apparently did so, too). The “Intercept” in our regression is the average processing time, and “Number.of.items” is the time to process each item. I used matrix math to calculate the wait times in our example, because I’m that awesome (or lazy, depending on your perspective).
The situation is complicated a bit by the method of payment: cash is by far the fastest (about 18 seconds) compared to credit card or check payments (41 and 54 seconds respectively). If you’re curious, here’s how I got those numbers in R:
> fit <- lm(seconds ~ Number.of.Items + Payment – 1, shopping,subset=-8)
> coef(summary(fit))
Estimate Std. Error t value Pr(>|t|)
Number.of.Items 2.955684 0.2997510 9.860464 6.340141e–11
Paymentcard 41.198885 6.7016800 6.147546 9.237407e–07
Paymentcard/cash 128.310215 22.1575721 5.790807 2.505366e–06
Paymentcash 17.974740 7.4723845 2.405489 2.252260e–02
Paymentcheck 53.997121 16.9930735 3.177596 3.430851e–03
Even if all the express lane patrons were paying with cash, the shorter line is still better even when the single patron pays with a credit card (104 seconds versus 97 seconds waiting).
