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Saturday, September 13, 2014

Queueing

You walk into a bank and see a single line for three tellers. There are four customers in line ahead of you. Bummed out? Perhaps, but let me explain why having a single line is better than three lines (one for each teller).

First, make a few assumptions. Customers walk into the bank one at a time, and each arrival is independent from the others. The time between two consecutive arrivals is random, but the probability of a long gap between arrivals is less than the probability of a short gap. These are attributes of what mathematicians call a Poisson process. The amount of time that it takes a teller to serve a customer is independent from one customer to the next. Again, the specific duration of each transaction differs, but most of the transactions are short whereas a lengthy transaction is the exception. These are attributes of what mathematicians call an exponential distribution.

What matters to you, the customer, is how long — on average — your stay inside the bank will be. Suppose 10 customers come into the bank every hour (on average) and that one teller can handle 20 customers per hour (on average). Mathematicians call this an M/M/1 queue, and there is an exact calculation for the expected value of how long you will stay in the bank: 6 minutes, including the time it takes for the teller to handle your own transaction. Remember, this is an average value.

Now suppose the bank is three times busier, with 30 customers arriving every hour. There are three tellers, but each of them still handles 20 customers per hour (on average). If there is a separate queue for each teller, the expected value of your stay inside the bank remains 6 minutes. But if there is one consolidated queue for all three tellers — mathematicians call this an M/M/3 queue — the expected value of your stay inside the bank falls to 3.47 minutes. That's right: one consolidated queue is far more efficient than separate queues. Moreover, if you are a statistician and you know what variance means, the variance of the waiting time is less for a consolidated queue than for separate queues. And lastly, one consolidated queue precludes the emotional problem when someone who entered the bank behind you reaches a teller before you do.

There is a handy online calculator that you can experiment with.

You see, I hope, why consolidated queues are what most banks and airlines use, as well as all amusement parks (aside from priority lanes). Unfortunately, most supermarkets and big-box retailers don't do what the mathematics say is best.

This comes from a branch of applied mathematics called queueing theory. I loved it in grad school. And as far as I know, "queueing" is the only word in the English language with five contiguous vowels.