From the time of birth (usually involving an approximately 9-month period from the moment of conception) until death (an entire life-time - whether brief, extensive or in between) and at many moments along the way human beings often find themselves waiting for things, events, conditions, etc. A major topic of Applied Mathematics that deals with this phenomenon of waiting is called Queuing Theory. Using the word "Queue", which is more common in British than American English and means "a line up" or "to form a line", a closely reasoned body of mathematical theory has been developed to describe this common human activity - theory applicable to normal economic activity. Realistic applications can be made to the phenomena of customers awaiting the delivery of goods/services, as well as to goods-in-process coming to be finished goods.
Queuing Theory arises from the use of powerful mathematical analysis to theoretically describe production processes along with statistical/probabilistic techniques to account for varying dynamic patterns within the stages of a productive process. The problem to be met - that occasioned the development of such theory - is simply entitled "congestion", what happens when a system does not operate smoothly or efficiently.
At this point the reader is invited to conjure up his/her own multiple general examples for the application of the theoretical concepts that follow. The demand for solutions to congestion problems arises all across the board of the international economy, as well as in the course of daily living. The origination of the formal study of Queuing Theory is credited to A. K. Erlang, a Danish telephone engineer who in the 1920's was attempting to predict telephone call service.
As mentioned above Queuing Theory examines the progress of customers (people) pursuing offered services, as well as goods-in- process (things) achieving the status of completed goods (whether capital or consumer goods). There are 3 areas of focus: Arrivals, Queue, Service Facility - each of which is further subdivided by a variety of analytic detail. A single process may consist of more than one stage/station, if the person/product passes through a series of Service Facilities; and it may also have more than one actual Queue/channel/ waiting-line leading to the subsequent Service Facility(ies). This paper will not include specific mathematical formulas or equations employed by Queuing Theory. But rather, it will attempt to portray analytic detail by listing a variety of brief definitions and descriptions.
The first area of focus considers Arrivals - entries into a productive system. (Please note that the theory applies equally to customers and goods-in-process). The number of Arrivals is significant (limited - few/many; or unlimited - possibly infinite). The pattern of arrival may admit of a tight/loose schedule (so many per time period) or completely at-random Arrivals. The behavior of the Arrivals may consist of quiet, patient people, stable/unstable objects, querulous children/adults, etc. Thus the first area of concern is the nature and number of the object(s) entering the process.
Secondly, attention is drawn to the Queue/Waiting-Line itself, which once established could have a limited (few or many) or unlimited population. Of greater importance is the Queue Discipline - the method by which a new Arrival advances to actually begin receiving service. The most common method is FIFO (first in, first out). But there are other methods: LIFO (last in, first out); pre-assigned priority (advance to service individually determined before arrival); priority by types (categories established before arrival for various reasons, e.g., length of service time <shorter required service time advances earlier> and size of economic cost for waiting <greater cost advances earlier>); and preemptive (new Arrival displaces another person/object in the Service Facility and the former returns to the Queue before continuing to receive the service). We also mention here that some customers may remove themselves from an active waiting line and so be "lost" to the system - if they are disgruntled (upset with waiting) or if they find that the good/service is out-of-stock/unavailable.
While it may or may not be actually calculable, there is at least an economic cost attributable to each person or good-in-process while remaining in the Queue. This is a practical consideration for a manager who is concerned with minimizing cost in the productive system.
Finally we come to the Service Facility as treated in Queuing Theory. Availability, whether the facility is free or already in- service, is of prime concern. Service Time is another critical issue, recognizing that this can vary - as a single server or several may provide the same service to individual customers/goods-in-service or as the individual needs of particular customers/goods-in-service differ as they come to the server. This could be the most unstable variable in non-production line processes. Capacity of the Service Facility, whether of one or more stations (number of customers/goods-in-process that can be serviced simultaneously), is another dimension of the analysis. By extension we note here that the total system - Queue(s) and Service Facility(ies) combined - may have a certain limited capacity, which would in turn restrict the number of new Arrivals to exactly matching customers/goods-in-service exiting the Service Facility.
The practical purpose of Queuing Theory is to provide examination tools for systems composed of Queues leading to Service Facilities, so that the systems may be made more efficient. Queuing Theory deals mathematically with both the regularities and irregularities of such systems - ultimately identifying occurrences of congestion (resulting from irregularities) and offering avenues for improving efficiency, as well as producing specific numerical data for further application. The measurements of congestion offered are the following: 1) the mean and distribution of time spent in a Queue; 2) the mean and distribution of customers/goods-in-service in both the Queue itself and the entire system; 3) the mean and distribution of the Service Facility's "busy periods" (utilization/unavailability). Each of the 3 distributions above can be expressed in terms of mathematical probabilities.
Queuing Theory with its fine-tuned analysis provides a base for a somewhat simplified and easier to use set of tools known as Model Building and Simulation. The practical production/operations manager recognizes that there is a trade-off between queuing costs and service costs and that together they make-up a total cost shared by the producer and the customer. With an eye toward reducing overall costs he/she will seek an optimum mix of queuing and service costs to achieve this goal.
Simulation from adequate Models is the technique that bridges the gap between the theoretical plane of Queuing Theory and the practical tool of a Decision Support System. Simulation is the experimental laboratory for testing changes in a productive system through the use of mathematical models - realistically incorporating the elements and patterns of a productive system while presenting key variables subject to experimental manipulation. Using a calculator or a computer, a researcher can insert appropriate values for these key variables into the model, run the simulation and derive achievable values for the actual system without intervening in the system itself. With a computer the model can be run quickly and repeatedly with various modifications.
POM textbooks draw information from Queuing Theory to underpin
their treatments of Model Building and Simulation. A case in point
comes from the Heizer and Render text - (Heizer, Jay and Barry Render.
Production & Operations Management, Upper Saddle River, NJ: Prentice
Hall, 1996, pp. 446-60). Under the heading of "Characteristics of a
Waiting-Line System", they spell out Arrival, Waiting-Line, and Service
Facility in a pattern similar to what has been given above. The authors
then proceed to describe 4 specific models as useful POM applications:
Queuing Theory and Simulation work hand-in-glove to uncover and smooth out some of the rough spots in a productive process - whether this involves delivering a service or a fabricated item to the immediate consumer. When in use a gloved hand reveals only the glove; but without the hand inside the glove is merely an empty shell. Simulation Models only work because of the analytic power of Queuing Theory which underlies and enables them.
Some interesting Queuing Theory Links:
http://web.calstatela.edu/faculty/hwarren/a503/queuingt.htm - Description Outline of Ququeing Theory an accounting class at Cal State, LA
http://www.lmet.fr/cgi-bin/Nouveaute/en/t/0-13-206111-2 - description of book titled PRACTICAL PLANNING FOR NETWORK GROWTH + DISK, Author : BLOMMERS; Editor : PRENTICE HALL; in which reference to queuing theory is made.
http://byte.com/art/9506/sec8/art9.htm - An article in BYTE which references queuting theory called Break Up Your Network
http://scad.utdallas.edu/scad/books/kalashnikov.html - Table of Contents for book titled MATHEMATICAL METHODS IN QUEUING THEORY, by V.V. Kalashnikov, of the Institute of Systems Analysis, Moscow
http://zenon.inria.fr/mistral/personnel/Nicolas.Niclausse/ref.html - References for computer technical uses of queues
http://www.ee.uwa.edu.au/~ccroft/eom446/leci.html - Basic description of both MRP and Queuing Theory Systems
http://forum.swarthmore.edu/maw/essay96.html - Theme essay written for Mathematics Awareness Week 1996 by Paul Davis, Worcester Polytechnic Institute
http://pitman.ma.rmit.edu.au:8080/Subjects/st824.html - Syllabus of Queuing Theory Class offered by ROYAL MELBOURNE INSTITUTE OF TECHNOLOGY, FACULTY OF APPLIED SCIENCE, DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCH