Queuing Theory In Detail

As noted yesterday, people lining up at amusement parks is a perfect example of queuing theory.

Queuing theory is defined as the mathematical study of waiting in lines and congestion.  It looks at every stage of waiting from arrival to how to arrange things to the end.

We see its application at amusement parks most of the time but its also applied at banks, grocery stores, airport security, or waiting on the phone for help.  Its also applied to computers and communications when queuing theory is applied to make sure information travels efficiently.

Queuing theory got it's start back in the early 1900's when Danish mathematician A. K Erlang explored the best number of circuits and switchboard operators needed to offer decent phone service for the Copenhagen Telephone Service. The resulting paper, published in 1909, proved that arrival's in queues could be modeled using the Poisson process.

In 1934, an American Engineer, figured out a mathematical framework for switching packets which allows information to travel the internet in modern society.  It is also responsible for keeping the movement of phone calls, streaming videos, etc.  Over the years it has been applied to traffic engineering and hospital emergency room management.

The mathematical description comes down to a queue with arrival time A, service time distribution B,  and servers C or the A/B/c/S/N/D queue.  The arrival time A actually looks at the time between arrivals and is subject to the Poisson distribution while the distribution B looks at the total number of items in the whole system including those in the queues.  S stands for the time it takes a customer to be serviced while c represents the number of servers in the system. N specifies the total number of customers, and D is the type of system such as first come first serviced or first in last out.

For a single server, this follows an exponential distribution but in real life, there are usually multiple servers so it gets a bit more complex.  This information is applied to call centers so that the number of operators handling the calls is set to make it as efficient as possible because the cost of operators is the most expensive part of running a call center.

On the other hand, Little's Law which states the average number of of items in the queue is found by multiplying the average rate at which items arrive in the queue by the average amount of time spent in the queue. It tends to be more generalized than the previous law meaning it can be applied to many systems regardless of the of the type of items or the way they are processed.

When you have more than one "server" you end up with networks that have to have queuing theory applied to.  We often see "networks" as we deplane airplanes, make our way to the next plane or to the baggage area, or checking in at ticket counters, move through security, and boarding airplanes.

With these types of activities, queuing network theory has developed.  There are two types of networks, those that are open, and those that are closed.   Open networks are defined as a system where customers can enter or leave at more than one place whereas a closed network has a fixed number of customers.  Closed networks include things like a set number of software licenses for computers where an airport is a great example of an open network.

So now you know more about queuing systems and their applications in real life to situations other than amusement parks.  Let me know what you think, I'd love to hear.  Have a great day.