beginners guide to queuing theory

The Beginner’s Guide to Queuing theory

The queuing theory studies and models the inner dynamics of queues, and ways in which lines could be managed more efficiently.

It is a massive topic, which includes many different facets of the waiting experience, such as:

  • Waiting behavior

  • Queuing and servicing models

  • Queuing disciplines

The definition of queuing theory

Queuing theory is the mathematical study of the formation and function of waiting lines. Queuing theory assesses the arrival process, service process, customer flow and other components of the waiting experience.

The application of queuing theory helps businesses improve the satisfaction of customers and employees, increase customer flow.

The history of queuing theory

The first paper on queuing theory was The Theory of Probabilities and Telephone Conversations by Agner Krarup Erlang, published in 1909.

Erlang worked with the Copenhagen Telephone Company and wanted to determine how many telephone circuits were necessary to prevent customers from waiting too long.

Erlang’s findings were applicable to many other fields, as telecommunications wasn’t the only industry suffering from long waiting times. This realization prompted him to develop his theory further into what later became known as the modern queuing theory.

Erlang’s role in developing the queuing theory has been recognized by naming the international unit for telephone traffic the erlang (E) in his honor.

A single cord circuit has the capacity to be used for 60 minutes in one hour. Whole 60 minutes of traffic constitute 1 erlang.

Why is queuing theory important?

Queues are a way of dealing with the imbalance between supply and demand. As long as demand (the number of customers) exceeds supply (capacity and the number of service points)

Queuing theory provides the tools and understanding for optimizing queues. As such, it has applications in many different industries, including but not limited to:

  • Retail

  • Telecommunications

  • Transportation

  • Finance and banking

  • Computing

  • Logistics

  • Project management

No matter the industry, as long as there is some waiting involved, businesses tend to bank on queuing theory.

Especially in customer-facing industries, managing waiting lines is becoming an increasingly bigger part of the experience package.

Because although not many customers admit to it, queues are among the biggest deciding factors when it comes to satisfaction. Waiting experiences can even impact the overall impression from the interaction with a business.

Understanding the underlying mechanism of queues gives businesses the opportunity to better prepare for customers and optimize their processes.

Understanding the queuing theory

When looking at queues through the lens of the queuing theory, it’s not enough to only talk about the length of the queue.

The queuing process can be broken down into four equally vital components:

  1. Arrival, which refers to the arrival of customers who are first in line.

  2. Capacity, which refers to the system’s limit with regards to the number of customers in line.

  3. Service, which refers to service points where service occurs.

  4. Departure or output, which refers to customers leaving the system after receiving service.

In turn, all of these aspects of queuing can be further broken down into smaller segments, i.e. arrival and departure rates, average service completion times, queuing discipline, number of servers, etc.

The system has one or more servers which manage customers from their arrival to their departure. The classic example of such a design is a cashier at a supermarket.

Little’s theorem

John Little came up with a law that describes the relationship between the distribution rate of customers and time spent by them in the system.

Little’s theorem can be summarized in this short equation:

L = λW

The average number of customers (L) is calculated from the average customer arrival rate (λ) multiplied by the average service time for a customer (W).

To understand this dynamic, consider your typical restaurant:

  1. If the arrival rate doubles but the service time remains constant, the number of customers will rise twofold.

  2. By the same logic, should the average service time double without the arrival rate changing, the number of customers will double, as well.

If customers arrive at the rate of 10 per hour and stay in the restaurant for an hour (1), the average numbers of customers at any time will be 10.

L = 10*1 = 10

Kendall’s notation

Queuing theory studies the behavior of single queues, also called queuing nodes. David George Kendall proposed a system for classifying these queuing nodes — the so-called Kendall’s notation.

According to Kendall’s notation, queuing nodes are described as A/S/c/K/N/D:

  • A for the arrival process.

  • S for the mathematical distribution of the service time.

  • c for the number of servers.

  • K for the capacity of the queue (if not unlimited).

  • N for the number of possible customers (if not infinite).

  • D for the queuing discipline (first-in, first-out by default).

Depending on the theoretical model used, the last three nodes can be omitted from the equation, making it a more manageable A/S/c.

Based on how each of the parameters is specified, we get different queuing models. Some of the more well-known models are M/M/1, M/M/c (also called Erlang-C model), M/G/1, M/D/1 and more.

These models deal with the mathematical theory of probability and are used to describe models of distribution in computation and logistics.

Unless you know what a Poisson process is, you shouldn’t worry about studying these models for your business.

In case you’re wondering: the M in the name refers to a Markov chain — a model describing a sequence of possible events whose probability depends on the state attained in a previous event.

In simpler terms, the future and past states of the system are independent. What happens tomorrow depends on today’s state and nothing else.

(Again, don’t bother learning that. It’s not necessary for serving customers.)

Queuing models

To determine the queuing model we’re dealing with, we need to look at two main parameters: the number of channels (or servers) and the number of phases of service. Think of channels as the number of stations where you receive service, and phases as the number of steps you need to get full service.

Each parameter can take two values: single (one), or multi (several). Different combinations of channels and phases give us four distinct types of queue management:

1. Single Channel, Single Phase

A single-channel, single-phase business has only one server. As soon as a customer is attended to, they receive full service.

Example: an automated car wash.

2. Single Channel, Multi Phase

A single-channel, multi-phase business has one server and a multi-step servicing process.

Example: retail banking, with different counters for withdrawals, deposits, new accounts, etc.

**3. Multi Channel, Single Phase **

A multi-channel, single-phase business has several servers and a one-step servicing process.

Example: airline ticket counter with separate queues for business class and economy class passengers.

4. Multi Channel, Multi Phase

A multi-channel, multi-phase business has several servers and a multi-step servicing process.

Example: a laundromat with several washers and dryers.

Read more: The Definitive Guide to Queue Management Systems

Queuing disciplines

A queuing discipline refers to the method of choosing which customers to serve in which order.

First-in, first-out queuing

A FIFO queue is a queue that operates on the first-in, first-out principle, hence the name. This is also referred to as the first-come, first-serve principle.

FIFO queuing refers to a queuing discipline where customers are served in the exact order in which they arrive. The first to join the line is the first one to leave it, all other factors being equal.

FIFO queuing is the predominant method of queuing, as it promotes fairness and its rules are universally understood. You can see it almost everywhere: banks, post offices, DMVs, retail, etc.

Read more: Understanding FIFO: First-In, First-Out in Queue Management

Last-in, first-out queuing

Although the principle of last-in, first-out sounds illogical at first, there might be some power to it.

Professor Lars Peter Osterdal of the University of Southern Denmark thinks that LIFO actually has the edge over FIFO: “The problem with a regular queue where you serve first those who arrive first is that people tend to arrive too early.”

LIFO forces people to come at staggered times, resulting in shorter queues. Coming early poses more risks, which is why in Osterdal’s experiment people chose to come later and spend as little time in the queue as possible.

Since this system rewarded late-comers, it wasn’t popular with people who were more accustomed to a traditional way of queuing.

There was nothing stopping some customers from leaving the queue and rejoining it to gain advantage, which had a negative effect on people trying to play by the rules.

That’s why LIFO is usually reserved for solving transportation and logistical problems, rather than issues of customers standing in line.

Priority queuing

In priority queuing, some customers have a special status which allows them to skip the usual means of queuing.

This type of queuing is most commonly seen in industries where there can be emergency cases — for example, healthcare. A patient with a severe case is naturally treated ahead of everyone else.

In non-healthcare industries, this type of queuing is usually called VIP queuing. For instance, business class passengers board the airplane before others.

The same goes for clubs, restaurants and similar venues.

The drawback of this system is that it may appear unfair unless it was clearly communicated in advance. After all, getting jumped ahead of you sounds infuriating if you weren’t aware that the business had priority queuing in place.

Queuing psychology and its impact on queuing theory

One of the problems of any theoretical model that tries to describe human behavior is that actions can appear illogical.

Queuing theory is helpful when it comes to explaining the benefits of a certain queuing flow, but it does ignore another important aspect of queuing: how customers feel while waiting.

Thankfully, there has been scientific research done in this area, as well. Queuing experts have determined that:

  1. Unoccupied time feels longer than occupied time.

  2. Unexplained waits are perceived as longer than they really are.

  3. Unfairness of the queue, be it real or perceived, impacts how customers feel.

To learn more about queuing behavior, its application and benefits to your business, refer to our guide to wait time psychology.


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