Aggravation in the Grocery Store: Modeling the Checkout Line Almost everyone has waited – likely impatiently – in a grocery store checkout line. The aggravation rivals another modern irritation – being stuck in traffic. And just like understanding traffic might ease the annoyance (see the reference box for two prior articles on traffic congestion), understanding the dynamics of cashier lines at grocery store might also give some mental relief.

So let’s explore.

The Need for More Cashiers

As we wait in line, we often wonder why the store doesn’t add more cashiers. The store must be trying to save money, at our expense and on our time.

However, our reaction doesn’t quite hit the mark. More cashiers will not fundamentally solve the waiting problem, nor does having less cashiers fundamentally save the store money. Why might the apparently obvious approach of adding cashiers not work? It might not work because the fundamental problem stems from the TIMING of the cashiers.

Let’s do some simple modeling to understand this. After that, we will add sophistication, and model more complex situations.

Simple Modeling: An Early Morning Scenario

Imagine a grocery store early on a Saturday. As the store opens, a small cadre of early risers enters. In this (relatively simple) situation, what waits might these shoppers experience?

Let’s put some numbers to the scenario, to enable calculations. We want the scenario simple enough to grasp it intuitively but still representative enough to mimic reality. Let’s use these assumptions.

  • 30 Shoppers
  • 15 items purchased per shopper
  • per item checkout time of three seconds (i.e. scanning, bagging, etc.)
  • A added per shopper checkout time of 45 seconds (i.e. payment, etc.)
  • Three cashiers on duty

As the store opens, the shoppers surge in and after a few minutes the first of the 30 shopper arrivers at the cashiers. From that point, we will assume a shopper arrives at the checkout lines every 30 seconds.

Will these shoppers need to wait? How long? How many of them? Let’s step through events to find out. When the first shopper arrives at the checkout line, that shopper will go without waiting to one of the three cashiers (i.e. all three are available). The second shopper arriving at the checkout line will see one cashier busy (with the first customer), but will see two cashiers with no line and go without waiting to one of them. Similarly, the third arriving shopper will see two cashiers busy, but the third cashier with no line and go there.

Now the fourth shopper arrives. To which line do they go? Well, we are now 90 seconds after the first shopper’s arrival (three shoppers later times the 30 second arrival interval). Will the cashier checking out the first shopper be available in time? Certainly. Checkout requires 90 seconds – 15 times 3 seconds, or 45 seconds, for the items plus 45 seconds more per shopper. So the first cashier has completed checkout for the first shopper when the fourth shopper arrives at checkout.

So the fourth shopper goes to the first cashier, without waiting. This sequence will continue, for example the second cashier will finish with the second shopper just as the fifth shopper arrives at the checkout line. Thus no shopper will experience a wait.

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