Queuing Theory

Queuing theory is a specific division of mathematics that focuses on studying waiting lines (queues) in cases where there is an excess of demand for a service as compared to the availability of the service. It gives a way of looking at and analyzing the behaviour of systems which encounter congestion as a normal occurrence: call centres, computer networks, transportation, etc.

By observing queue length, customers’ waiting time, and server utilization, queuing models can become immensely beneficial in resource management and enhancement of systems performance.

In this article, we have covered the basics of Queueing Theory.

Queuing theory is a branch of mathematics used to describe, analyze and predict the length of the queues and waiting time in the system. Implies the development of models for the arrival process, service process, and queue discipline of the model. These models are then quantified, and the resulting probability theory and stochastic processes are used to calculate other performance criteria including the mean number of customers the system contains, the average time that a customer spends waiting in the system and the probability of a customer to wait for service.

The underlying assumption of queuing theory is that arrivals to the system are characterized by a probability distribution, the Poisson distribution, and service times by another known distribution, the exponential distribution. These assumptions enable analysts to devise easily solvable mathematical models, which may be used to evaluate system performance.

Queuing theory works by modeling a system as a series of components:

• Arrival Process: The way customers arrive at the system, which can be described by an arrival rate and a probability distribution.

• Queue: The waiting line where customers wait for service.

• Service Process: A specific way of customers being served that may be quantified using a service rate and the probability distribution.

• Service Discipline: The system of serving the customers such as first come first serve (FCFS) or the prioritized system.

For example, consider a simple queuing system with a single server and a first-come-first-served (FCFS) service discipline.

Let’s assume that customers arrive according to a Poisson process with rate λ and that service times follow an exponential distribution with rate μ. The average number of customers in the system (L) and the average waiting time in the queue (W_q) can be calculated using the following formulas:

L = ρ / (1 – ρ)

W_q = ρ / (μ – λ)

where,

ρ = λ / μ is the utilization factor, which represents the fraction of time the server is busy.

Queuing theory originated in 1909 by Danish mathematician A. K. Erlang in the year 1909 while he was working with the Copenhagen Telephone Company. Erlang investigated the congestion of incoming calls in telephone traffic, and to evaluate the probability of calls arriving at the switchboard, he established mathematical models. He also provided the much-needed initial foundation for the growth of what would later become known as queuing theory.

Many other researchers have also added their findings to the development of queuing theory since then, including Agner Krarup Erlang (A. K. Erlang’s son), David Kendall, and John Little among others. Today, queuing theory has grown to be a popular and respected branch of study that is frequently used in operations research, computer engineering, and industrial engineering.

The basic elements of queuing theory are:

• Arrival Process: The way customers arrive at the system, which can be described by an arrival rate and a probability distribution.

• Service Process: The service aspect can be constructed from a service rate and a probability distribution for customers who are served.

• Queue Discipline: The method used to serve the customers or the order in which the customers are served such as the first come first served (FCFS) or priority-based.

• Number of Servers: The number of servers available to serve the customers.

• System Capacity: Limit of customers who may attend the system at a particular time including those being served and those waiting for their turn.

• Performance Measures: Quantitative measures that are used to measure the performance of the system like the queue length, waiting time, and server utilization.

To use queuing theory, you need to follow these steps:

• Define the System: Some of the elements that must be defined include the arrival pattern; the services offered; and the queue discipline.

• Collect Data: Collect data on the arrival rates, the service times and the number of customers in the queue with the aim of obtaining estimates of the parameters of the various queuing models.

• Choose a Queuing Model: Choose a queuing model suited to some characteristics of the system let them be the number of servers, service discipline, and arrival and services distributions.

• Analyze the Model: When selecting one of the queuing models, utilize mathematical methods/tools or simulation methods to compute performance metrics including waiting time and queue length.

• Interpret the Results: Queue analysis results must be used to make decisions about resource allocation, staffing levels, or system design.

Various real-life examples where Queuing theory is uses are:

Call Center Example: Consider a call center with a single server that receives calls according to a Poisson process with a rate of 10 calls per minute. The service time for each call follows an exponential distribution with a mean of 3 minutes. Using queuing theory, we can calculate the average number of calls waiting in the queue (L_q) and the average waiting time in the queue (W_q) as:

L_q = λ² / (μ × (μ – λ))

= 10² / (1/3 × (1/3 – 10))

= 10 calls

W_q = L_q / λ

= 10 / 10 = 1 minute

Supermarket Checkout Example: A supermarket has two checkout counters, each with its own queue. Customers arrive according to a Poisson process with a rate of 20 customers per minute and are served in a first-come-first-served (FCFS) manner. The service time for each customer follows an exponential distribution with a mean of 2 minutes. Using queuing theory, we can calculate the average number of customers in the system (L) and the average waiting time in the queue (W_q) as:

L = (λ / μ) / (1 – (λ / (2 × μ)))

= (20 / 30) / (1 – (20 / (2 × 30)))

= 1.5 customers

W_q = L / λ – 1/μ

= 1.5 / 20 – 1/30

= 0.075 minutes

Queuing theory has a wide range of applications in various fields:

• Call centers: For example, queuing models are applied to determine the optimal number of staff, likely waits for patients and ways to enhance customer satisfaction.

• Computer systems: It has its application in the study of the performance of computer networks, CPU scheduling, and memory management.

• Manufacturing: Through queuing models, one can optimize schedule production, reduce WIP (work in progress) and increase throughput.

• Transportation: It helps to analyze the traffic density, make efficient the flow of people in the airport and design the public transportation system.

• Healthcare: It is noteworthy that queuing models are employed to optimize patient flow, schedule appointments, as well as allocate facilities and manage hospital resources.

What is Queuing Theory?

Queuing theory studies the behavior of waiting lines to understand and optimize the flow of customers or items through a process.

What are the Four Assumptions of Simple Queuing Models?

Simple queuing models assume: that customers arrive at random, service times are random, there’s a single queue, and customers are served on a first-come, first-served basis.

How can Queuing Theory be applied to Real-Life Situations?

Queuing theory can optimize service in places like banks, hospitals, and supermarkets by reducing wait times and improving service efficiency.

How does Queuing Theory help Decision-Making?

It helps managers make informed decisions on resource allocation, staffing levels, and process improvements to enhance service and reduce costs.

What are Advantages of Queue?

Queues help organize service orders, ensure fairness, manage customer expectations, and improve overall service efficiency.