Queueing Theory


The British Standards Institution’s definition is: “The mathematical theory used in the analysis of systems in which service is provided under conditions of varying supply and demand”, Term number 21064.

More comprehensively the subject is concerned with the mathematical analysis and solution of problems in which items requiring service or items providing service occasionally stand idle, and in which it is required to specify either the arrival rate or the service rate or both, to achieve the most satisfactory style of operation, e.g. least cost operation of the total system.


Mathematical techniques, including probability theory, used in operations or operational research to identify, illustrate and, it is hoped, influence the characteristics of queues, whether of people, materials, work-in-progress, etc. The object is to find the best way of planning the sequence of events so that bottlenecks can be avoided.


Queueing theory deals with problems of congestion in which two opposing pressures must be balanced. These two pressures are the rate of joining the queue and the rate of discharge from it. Solutions can be reached analytically and in some instances by alternative methods (simulation, qv). The theory of queues requires a long exposition in which somewhat complex mathematics are used. There can be no brief treatment of the subject that will handle real cases. The analytical treatment depends on the probability distribution used to represent arrival and service rates and the queue discipline.


One of the simplest examples that could be chosen to illustrate queueing theory is the case of issuing stores. Ideally, the two requirements are that the storemen will never be idle for lack of customers and that the customers will never have to wait. These two requirements are mutually exclusive – the more one approaches 100 percent utilisation of storemen, the greater the waiting time of the customers. On the other hand, in order to guarantee that customers will never wait, the number of storemen must be increased to infinity. Both classes of idleness involve costs. The object of queueing theory is usually to minimise the overall cost.

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