Queuing Theory

Queuing Theory is a field of mathematics concerned with queues (i.e. waiting lines). It is considered to be a branch of operations research, since valuable business decisions can be made based on the inputs from queuing theory. The fundamental problem which queuing theory attempts to solve is the prediction of the waiting line length and waiting time using a model-based approach. Thus, it can be also stated that it is concerned with explaining the queue formation and propagation. Queuing theory is closely related to probabilistic theory and stochastic processes in terms of using its valuable mathematical tools. On the other hand, methods from queuing theory are applicable in numerous disciplines: communications, operations research, reliability engineering, microprocessor electronics, management & business organization, computing, traffic engineering, psychology, etc. You can probably think of many other real-world applications where knowing how a waiting line will change is an advantage. Whether we wait for service to be performed (e.g. supermarkets, banks), wait for public transport to arrive (bus, subway), or any kind of response from a machine (computers or smartphones), queuing is part of our everyday experience.

 Some of the topics covered in a standard course on queuing theory would be:

• Topics from probability and stochastic processes relevant for queuing theory (random variables, random processes, types of distributions, moments, central limit theorem, law of large numbers, etc)
• Performance measures of queuing systems
• Description of queuing systems using Kendall’s notation
• Infinite- and finite-source systems
• Analysis of scheduling policies (first-come-first-served, first-come-last-served, served-in-random-order, processor sharing, shortest job first, etc)
• Birth-death processes and simple Markovian queues
• Multidimensional birth-death processes
• Single- and multi-server queues
• Queuing networks (Jackson networks, BCMP theorem)
• Little’s law
• Asymptotic analysis of queues
• Programming simulations of queuing models
• Renewal theory

To understand the basics of Queuing Theory, it is important to first have a brief overview of the standard notation:

• λ: average rate of arrival of customer
• µ: average service rate
• ρ = λ / µ for single server queues: the probability that the server is busy or the probability that someone is being served
• s: number of servers
• P_n: probability that there are n customers in the system
• L: average number of the customers present in the system

Additionally, models are described via Kendall’s notation as follows:

A / B / s / q / c / p

where:

A stands for the distribution describing arrival process,

B stands for the distribution describing service (distribution of service duration),

s stands for the total number of servers,

q stands for the queuing discipline (FIFO, LIFO, SIRO...)

c stands for the system capacity in case of a finite queue, and

p stands for the population size (number of possible customers).

When omitted, common practice is to assume that q = FIFO (First-In-First-Out), c = ∞ (i.e. infinite queue), and p = ∞ (i.e. open system). A typical example of a queuing model is M/M/1: M stands for Markovian process in both arrival and service process, with single (1) server.

Apart from your study materials and classes, you can find many useful lectures and various explanations concerning queuing theory on these websites:

• MIT open course - Queues: Theory and Applications
• Introduction to Queuing Theory from Florida Atlantic University by Dr. Bob Cooper: lecture videos and book
• Basic Queuing Theory, Dr. Janos Sztrik, University of Debrecen (book)
• Basic relations in Queuing Theory from Missouri University of Science and Technology
• Fundamental formulas in Queuing Theory from KTH Royal Institute of Technology (Sweden)

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