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The Science Behind Your Long Wait in Line by: Jo Craven McGinty

Oct 08, 2016

Click here to view the full article on WSJ.com

You’ve probably participated in this familiar dance: Given a choice of checkout lines, you’ve

somehow picked the slowest.

You could wait it out. You could chassé to another queue. Or you could bail out altogether. After

all, no one likes to wait. But are the other lines really faster?

When parallel lines feed multiple cashiers, you may not be in the slowest one, but chances are,

you also are not in the fastest.

Bill Hammack, a professor at the University of Illinois at Urbana-Champaign and YouTube’s

“Engineer Guy,” explained it like this:

Imagine three lines feeding three cash registers. Some shoppers will have more items than others,

or there may be a delay for something like a price check. The rate of service in the different lines

will tend to vary. If the delays are random, there are six ways three lines could be ordered from

fastest to slowest—1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2 or 3-2-1. Any one of the three (including the

one you picked) is quickest in only two of the permutations, or one-third of the time.

There are two sources of variability, according to Linda V. Green, a professor of health-care

management at the Columbia University Business School who specializes in mathematical

models of service systems.

“When the demands for service come in and how long it takes the servers to process them,” she

said. “That inevitably causes temporary mismatches between supply and demand and hence

backups, delays and congestion.”

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Luckily, most service providers take steps to manage the wait. A supermarket with parallel lines

reserves some registers for customers with fewer items. Airline security uses a single serpentine

line to feed multiple agents to mitigate bottlenecks at individual checkpoints. Emergency rooms

and 911 dispatchers give priority to those whose needs are most urgent.

Each approach is based on queuing theory, or the mathematical study of lines.

Queues can be trivial, like a line at an ATM, or they can be serious, like a list of people waiting

for an organ transplant, said Richard Larson, director of Massachusetts Institute of Technology’s

Center for Engineering Systems and an expert on queuing, but the fundamentals are the same: A

basic queue funnels clients demanding service to one or more servers who respond. If the servers

are busy, other demands must wait.

The clients may include a line of people, a series of 911 calls, or a string of commands issued

over a computer network (think of a printer queue). The servers are the cashiers, the dispatchers

or the devices that respond.

Queuing theory helps untangle the mess of requests, or at least smooth it out, by estimating the

number of servers needed to meet demand over a given period and designing rules for advancing

the queue.

The best system depends on the situation. “First come, first served” is most familiar, and people

often prefer it because it seems fair. But most also accept that a heart attack should take

precedence over a sprained ankle or someone with five items shouldn’t have to wait behind a

procession of brimming shopping carts.

Queuing theory originated in 1908 when a Danish scientist named Agner Krarup Erlang went to

work for the Copenhagen Telephone Company and set about trying to determine how many

telephone trunks, or lines, were needed to serve callers.

The company could have provided a trunk for each telephone, Dr. Hammack said, but that would

have been wasteful because not everyone calls at the same time. It could have provided enough

trunks to handle the average number of calls, but too many would be blocked when the average

was exceeded.

“Think of it like a bike wheel,” said Dr. Larson. “The switchboard is the hub. Each spoke is a

wire coming in from a home to a human operator who can connect your wire to some other

spoke in the wheel. What should the capacity be? If it’s too big, you’ll spend too much money on

capital investments. If it’s too small customers won’t be able to be connected reliably.”

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Customers waited in line to check out at an Ikea store in New York City in September. PHOTO: MICHAEL NAGLE/BLOOMBERG NEWS

Erlang’s basic formula included three parameters: the number of trunks, the number of calls per

hour and their average length. Adjusting the number of trunks in the formula raised or lowered

the probability that a call would be blocked during the busiest hour.

Since then, retail stores, banks, call centers, emergency rooms, manufacturing plants, computer

networks and all variety of queuing environments have used Erlang’s formulas or related models

to figure out how to manage their lines.

Meanwhile, clients—at least the human variety—can’t help but wonder if there isn’t a way to

game the system.

Raj Jain, a computer-science and engineering professor at Washington University in St. Louis

takes an analytical approach: When he’s in line, he times the service. “Then I count the number

of people ahead of me, and I know how much time I am to wait.”

Dr. Larson takes a different approach. “I just strike up a conversation with an adjacent queue

dweller,” he said, “and wait.”

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Appendix:

Erlang B & Erlang B Formula

- details & tutorial describing the Erlang B unit, and Erlang B formula developed from the basic

Erlang unit to handle blocked calls into the calculations.

ERLANG TUTORIAL INCLUDES

 Erlang telecommunications tutorial

 Erlang B

 Erlang C

The Erlang B is used to work out how many lines are required from a knowledge of the traffic figure during the busiest hour. The Erlang B figure assumes that any blocked calls are cleared immediately. This is the most commonly used figure to be used in any telecommunications capacity calculations.

Erlang B

It is particularly important to understand the traffic volumes at peak times of the day. Telecommunications traffic, like many other commodities, varies over the course of the day, and also the week. It is therefore necessary to understand the telecommunications traffic at the peak times of the day and to be able to determine the acceptable level of service required. The Erlang B figure is designed to handle the peak or busy periods and to determine the level of service required in these periods.

Essentially, the Erlang B traffic model is used by telephone system designers to estimate the number of lines required for PSTN connections or private wire connections. The three variables involved are Busy Hour Traffic (BHT), Blocking and Lines.

Busy Hour Traffic (in Erlangs) is the number of hours of call traffic there are during the busiest hour of operation of a telephone system.

Blocking is the failure of calls due to an insufficient number of lines being available. E.g. 0.03 mean 3 calls blocked per 100 calls attempted.

Lines is the number of lines in a trunk group.

The Extended Erlang B is similar to Erlang B, but it can be used to factor in the number of calls that are blocked and immediately tried again.

Erlang B formula

Where: B=Erlang B loss probability N=Number of trunks in full availability group A=Traffic offered to group in Erlangs