Mathematical methods in queueing theory
Edwards, Jane Joan
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The thesis deals with some of the mathematical techniques that are used in solving queueing theory problems. The organization of the paper is basically the same as that used by Goddard  in his treatment of queueing theory. The first method investigated is the analysis of queueing problems as Markovian processes. This analysis is due to D. G. Kendall [4,5] and is primarily for the case M/G/1. Formulas are found for E(n) and E(w). The limitations of this method in dealing with more general queues are mentioned. For the case M/M/s, the differential-difference equations are developed with examples of their use in machine breakdown problems and telephone trunk line congestion. The treatment is primarily for the case in which the system is in statistical equilibrium. The uses of Laplace transforms and probability generating functions are illustrated by Pollazoek's method of finding the moments of the waiting time distribution for the queue M/G/1. They are also shown in finding Pn(t) and Pn in an example of welders using a power supply. A method of expressing the distribution of the waiting time for the queue G/G/1, due to Lindley , is outlined at the conclusion of the paper. The result is an expression for the waiting time distribution in terms of the density function of the difference between the service time and inter-arrival time, rather than either density function separately.
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