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# Some Problems in the Theory of Queues

David G. Kendall
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 13, No. 2 (1951), pp. 151-185
Published by: Wiley for the Royal Statistical Society
Stable URL: http://www.jstor.org/stable/2984059
Page Count: 35
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## Abstract

The paper opens with a general review of some points in congestion theory, and continues with a simplified account of the Pollaczek-Khintchine "equilibrium" theory for the single-counter queue fed by an input of the Poisson type and associated with a general service-time distribution. It is pointed out that although the stochastic process describing the fluctuations in queue-size is not (in general) Markovian, it is possible to work instead with an enumerable Markov chain if attention is directed to the epochs at which individual customers depart (these epochs forming a sequence of regeneration points). The ergodic properties of the chain are investigated with the aid of Feller's theory of recurrent events; it is found to be irreducible and aperiodic, and the further classification of the states depends on the value of the traffic intensity, ρ (measured in erlangs). When $\rho < 1$ the states are all ergodic and the system ultimately settles down to a regime of statistical equilibrium, the transition-probabilities converging to the values given by the equilibrium solution. When $\rho > 1$ the states are all transient, and when ρ has the critical value of unity the states are all recurrent-and-null. The paper closes with some further general comments, with a re-interpretation of Borel's theory of "busy periods" in terms of the Galton-Watson "branching process" of stochastic population theory and with a sketch of an argument leading to the distribution of the length in time of a busy period.

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