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Mathematics of Lines:
A Public Radio Commentary
by Bill Hammack
Listen using RealAudio
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
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