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Mathematics of Lines

December 21, 2004

A public radio commentary

I'm sure that this week you've been trapped in a slow moving line, likely for last minute gift shopping, or to return something. As an engineer, I have some good news for you about making those lines move faster. It comes from a branch of engineering called "queuing theory."

It began in 1909 at the Copenhagen Telephone Company when an engineer there, Agner Krarup Erlang, figured out how to direct telephone calls to unused phone lines. This is exactly the same as a check out line. You showing up to check out is like a phone call arriving, and an available cashier is like an open telephone line. Here's what Erlang learned that is relevant to your life.

To keep the lines moving it would seem that the store should just measure the number of people arriving in a typical hour and then assign enough cashiers so that usually everyone will be served quickly. Erlang showed that this was a recipe for checkout line gridlock.

He learned that people are as likely to arrive at one time as another, but precisely when they arrive is random. This means that people will arrive in bunches, not spaced out evenly. So, if stores have just the right number of cashiers for the average number of shoppers in an hour, the store will, at times, have too few cashiers, resulting in long waits. The solution is to combine the separate lines into one huge line, and let that line feed to more than one cashier. You see this used at airports and in banks. It works because there is a random chance you'll be behind someone who causes a delay: Someone, say, who needs a price check. If that delay happens when you're in a line for a single cashier you'll be delayed, but imagine if you were in that one line that fed, say, three cashiers. The only time you'd be delayed is when people in front of all three cashiers are delayed. This makes the single line about three times faster than having one line per cashier.

Erlang's work also answers the pressing question for this season: Why do other lines always move faster than yours? The answer: Because it's true. It's that randomness of delays again.

Picture yourself in a line, with a line on each side, so three lines total. If it's random that someone in one line will have trouble, all three lines are equally likely to suffer a delay, which means that there is only a one in three chance that your line will suffer the least. To put it another way: The chances are greater, two in three, that one of the other lines will move faster.

So, here is what mathematics is telling us: Sure, today that other line is moving faster, but some days you'll be in the faster lane. In other words, during this season: Let the odds be with you.

Copyright 2004 William S. Hammack Enterprises