# Queueing theory

Queueing theory is the mathematical study of waiting lines (or queues). The theory enables mathematical analysis of several related processes, including arriving at the (back of the) queue, waiting in the queue (essentially a storage process), and being served by the server(s) at the front of the queue. The theory permits the derivation and calculation of several performance measures including the average waiting time in the queue or the system, the expected number waiting or receiving service and the probability of encountering the system in certain states, such as empty, full, having an available server or having to wait a certain time to be served.

## Overview

The word queue comes, via French, from the Latin cauda, meaning tail. Most researchers in the field prefer the spelling "queueing" over "queuing",[1] although the latter is somewhat more common in other contexts.

Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide service. It is applicable in a wide variety of situations that may be encountered in business, commerce, industry, healthcare,[2] public service and engineering. Applications are frequently encountered in customer service situations as well as transport and telecommunication. Queueing theory is directly applicable to intelligent transportation systems, call centers, PABXs, networks, telecommunications, server queueing, mainframe computer queueing of telecommunications terminals, advanced telecommunications systems, and traffic flow.

Notation for describing the characteristics of a queueing model was first suggested by David G. Kendall in 1953. Kendall's notation introduced an A/B/C queueing notation that can be found in all standard modern works on queueing theory, for example, Tijms.[3]

The A/B/C notation designates a queueing system having A as interarrival time distribution, B as service time distribution, and C as number of servers. For example, "G/D/1" would indicate a General (may be anything) arrival process, a Deterministic (constant time) service process and a single server. More details on this notation are given in the article about queueing models.

## History

Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on queueing theory in 1909.[4]

David G. Kendall introduced an A/B/C queueing notation in 1953. Important work on queueing theory used in modern packet switching networks was performed in the early 1960s by Leonard Kleinrock.

## Application to telephony

The public switched telephone network (PSTN) is designed to accommodate the offered traffic intensity with only a small loss. The performance of loss systems is quantified by their grade of service, driven by the assumption that if sufficient capacity is not available, the call is refused and lost.[5] Alternatively, overflow systems make use of alternative routes to divert calls via different paths — even these systems have a finite traffic carrying capacity.[5]

However, the use of queueing in PSTNs allows the systems to queue their customers' requests until free resources become available. This means that if traffic intensity levels exceed available capacity, customers' calls are here no longer lost; they instead wait until they can be served.[6] This method is used in queueing customers for the next available operator.

A queueing discipline determines the manner in which the exchange handles calls from customers.[6] It defines the way they will be served, the order in which they are served, and the way in which resources are divided between the customers.[6][7] Here are details of four queueing disciplines:

First in first out
This principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.[7]
Last in first out
This principle also serves customers one at a time, however the customer with the shortest waiting time will be served first.[7]
Processor sharing
Customers are served equally. Network capacity is shared between customers and they all effectively experience the same delay.[7]
Priority
Customers with high priority are served first.[7]

Queueing is handled by control processes within exchanges, which can be modelled using state equations.[6][7] Queueing systems use a particular form of state equations known as Markov chains which model the system in each state.[6] Incoming traffic to these systems is modelled via a Poisson distribution and is subject to Erlang’s queueing theory assumptions viz.[5]

• Pure-chance traffic – Call arrivals and departures are random and independent events.[5]
• Statistical equilibrium – Probabilities within the system do not change.[5]
• Full availability – All incoming traffic can be routed to any other customer within the network.[5]
• Congestion is cleared as soon as servers are free.[5]

Classic queueing theory involves complex calculations to determine call waiting time, service time, server utilisation and many other metrics which are used to measure queueing performance.[6][7]

## Queueing networks

Networks of queues are systems which contain an arbitrary, but finite, number m of queues. Customers, sometimes of different classes,[8] travel through the network and are served at the nodes. The state of a network can be described by a vector $\scriptstyle{(k_1,k_2,\ldots,k_m)}$, where ki is the number of customers at queue i. In open networks, customers can join and leave the system, whereas in closed networks the total number of customers within the system remains fixed.

The first significant result in the area was Jackson networks, for which an efficient product form equilibrium distribution exists.

### Applications

Queueing networks have been applied to reduce waiting times in hospitals.[9]

## Role of Poisson process, exponential distributions

A useful queueing model both (a) represents a real-life system with sufficient accuracy and (b) is analytically tractable. A queueing model based on the Poisson process and its companion exponential probability distribution often meets these two requirements. A Poisson process models random events (such as a customer arrival, a request for action from a web server, or the completion of the actions requested of a web server) as emanating from a memoryless process. That is, the length of the time interval from the current time to the occurrence of the next event does not depend upon the time of occurrence of the last event. In the Poisson probability distribution, the observer records the number of events that occur in a time interval of fixed length. In the (negative) exponential probability distribution, the observer records the length of the time interval between consecutive events. In both, the underlying physical process is memoryless.

Models based on the Poisson process often respond to inputs from the environment in a manner that mimics the response of the system being modeled to those same inputs. The analytically tractable models that result yield both information about the system being modeled and the form of their solution. Even a queueing model based on the Poisson process that does a relatively poor job of mimicking detailed system performance can be useful. The fact that such models often give "worst-case" scenario evaluations appeals to system designers who prefer to include a safety factor in their designs. Also, the form of the solution of models based on the Poisson process often provides insight into the form of the solution to a queueing problem whose detailed behavior is poorly mimicked. As a result, queueing models are frequently modeled as Poisson processes through the use of the exponential distribution.

## Limitations of mathematical approach

Classic queueing theory is often too mathematically restrictive to be able to model all real-world situations exactly. This restriction arises because the underlying assumptions of the theory do not always hold in the real world.

For example; the mathematical models often assume infinite numbers of customers, infinite queue capacity, or no bounds on inter-arrival or service times, when it is quite apparent that these bounds must exist in reality. Often, although the bounds do exist, they can be safely ignored because the differences between the real-world and theory is not statistically significant, as the probability that such boundary situations might occur is remote compared to the expected normal situation. In other cases the theoretical solution may either prove intractable or insufficiently informative to be useful.

Alternative means of analysis have thus been devised in order to provide some insight into problems which do not fall under the mathematical scope of queueing theory, though they are often scenario-specific since they generally consist of computer simulations and/or of analysis of experimental data. See network traffic simulation.

## References

1. ^ wiktionary:queueing
2. ^ Mayhew, Les; Smith, David (December 2006). "Using queuing theory to analyse completion times in accident and emergency departments in the light of the Government 4-hour target". Cass Business School. Retrieved on 2008-05-20.
3. ^ Tijms, H.C, Algorithmic Analysis of Queues", Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003
4. ^ http://pass.maths.org.uk/issue2/erlang/index.html
5. ^ a b c d e f g Flood, J.E. Telecommunications Switching, Traffic and Networks, Chapter 4: Telecommunications Traffic, New York: Prentice-Hall, 1998.
6. ^ a b c d e f Bose S.J., Chapter 1 - An Introduction to Queueing Systems, Kluwer/Plenum Publishers, 2002.
7. ^ a b c d e f g Penttinen A., Chapter 8 – Queueing Systems, Lecture Notes: S-38.145 - Introduction to Teletraffic Theory.
8. ^ F. P. Kelly Networks of Queues with Customers of Different Types Journal of Applied Probability, Vol. 12, No. 3 (Sep., 1975), pp. 542-554
9. ^ Kira L., Schlechter (Monday March 02, 2009), "Hershey Medical Center to open redesigned emergency room", The Patriot-News