Queue or Queues

I am spending this year visiting the University of Auckland. New Zealand is a wonderful, beautiful country filled with very friendly people (see our personal blog for pictures and stories). It is not, however, a particularly efficient country. Service standards are not quite up to US standards. This is made worse by our living on an island. We love living on Waiheke, but any outing often results in “island efficiency”, which tests our relaxed, island mentality to its fullest. For instance, last night we went to a costume ball (medieval style). Beautiful converted barn, wonderful food, perfect music. And exactly one porta-potty for 200 guests at this four hour affair. Island efficiency.

Of course, operations research is about making things work better. And to an OR person, there is nothing more frustrating than seeing things work inefficiently. Grocery stores really stress me. I will not go to a busy store, since the lines and hassles will eventually give me a stroke. Instead, I shop at the off-off times whenever possible.

Some of this annoyance is completely avoidable, however, using just a bit of OR. In queueing, there are a few basic ideas. Here are a couple that even a two hour introduction to queueing should get to:

  • Assuming variance in either service or arrivals, if the service rate equals the arrival rate, the queue will explode.
  • All else being equal, one line leading to a bunch of servers is better for the customer than a bunch of lines each leading to one server.

So why is it that (almost) every grocery store has individual lines? Grocery stores are some of the most sophisticated users of techniques like data mining and logistics optimization, so they aren’t or-phobic. Why can’t they get this right?

One company does get this right, and that is Whole Foods, which gets a whole lot of things right in its operation (why else would we enthusiastically pay their prices?). The New York Times has an article entitled “A Long Line for a Shorter Wait” (thanks to Brian Borchers for pointing this out to me):

For its first stores here [New York City], Whole Foods, the gourmet supermarket, directs customers to form serpentine single lines that feed into a passel of cash registers. Banks have used a similar system for decades. But supermarkets, fearing a long line will scare off shoppers, have generally favored the one-line-per-register system. By 7 p.m. on a weeknight, the lines at each of the four Whole Foods stores in Manhattan can be 50 deep, but they zip along faster than most lines with 10 shoppers. Because people stand in the same line, waiting for a register to become available, there are no “slow” lines, delayed by a coupon-counting customer or languid cashier.

Now there is the psychological effect to avoid: the lines look long! So Whole Foods added some bells and whistles:

Perhaps the most important role players in the Whole Foods system are the “line managers,” who monitor the flow of people, direct them to a cash register and, when needed, hold up signs saying how long it will take to check out. In another innovation, color-coded digital screens are now replacing those humans.

To give an idea of how effective the single line is, suppose you have a single queue with 20 customers arriving per hour. If the cashier can handle (on average) 22 customers per hour (close to saturation, but probably roughly what “efficient” managers would aim for), then the queue will grow so long that the average wait will be 27 minutes! Five such queues would end up with about 50 people waiting in line on average. If you go over to one line (with 100 arrivals/hour) being served by five cashiers, the average wait goes down to under 5 minutes, and the number of people waiting in line is only 12 on average. Thanks to Armann Ingolfsson for putting together the Queueing Toolpack that makes these calculations easy.

There are a lot of assumptions in this analysis, and you can play around with input and service distributions, reneging, balking, queue hoping, and so on, but the result is robust to any of this: a single line to multiple queues is better than a bunch of single lines (about the only interesting aspect of multiple queues is the ability to have an express line for customers with just a few items). Every OR person knows this, and Whole Foods uses it. Every grocery store should.

I think it goes without saying that on Waiheke island, every line is a bunch of lines each being served (or not) by a single server. Even in banks.

4 thoughts on “Queue or Queues”

  1. By the way, one more thought. The single-line queue does take a lot of discipline on the part of “efficiency” minded managers. Such a manager might look at the single line going in to 5 cashiers with a short wait and short queue and decide he could save money by getting rid of a cashier. Terrible idea here: four cashiers can’t handle the load, and the queue will increase unboundedly. Perhaps single-line queues hide instability a bit more than multiple single-line queues (the long lines make it clear that 5 cashiers are barely enough).

  2. Newzealand is a beautiful country….I would definitely love to go there and work as a cashier at the grocery store….

  3. I think that the largest hurdle in grocers migrating to a single-line, multiple checkout system lies in making the transition. If it took you an entire post to explain the advantages of this system to an OR-savvy (and, of course, captive) audience, how can a grocer explain it to the average Joe (or Jane) quickly, efficiently, and effectively such that the immediate psychological barriers are overcome? How can you sell this idea to consumers at a single glance?
    Such a proposal would revolutionize the grocery (or similar retail) industry – just think of the changes to the store layout, or the chances for interacting with the customers in the queue to enhance their visit or promote your chain, vendor, etc. The entire shopping experience would change to a degree. And, of course, the impulse buys of Snickers, tabloids, and flashlight batteries would take a hit…
    No retailer would want to be the first to try a new and “unproven” system knowing the startup risk to buisness (“Hey, how come y’all only gots 1 line in this here store?” repeated 1000X over) given that if it is successful, all other competitors will follow suit. What will you have gained? Why prime the market when all others can reap the benefit after you have shouldered the risk?

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